CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Group (mathematics) - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy","wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"b636a9eb-3474-4d44-8576-7ca8c6fe4dd5","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Group_(mathematics)","wgTitle":"Group (mathematics)","wgCurRevisionId":1280089198,"wgRevisionId":1280089198,"wgArticleId":19447,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Use shortened footnotes from September 2024","Articles containing French-language text","CS1 French-language sources (fr)","CS1 German-language sources (de)","Articles containing German-language text","CS1: long volume value","Featured articles","Group theory","Algebraic structures","Symmetry"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Group_(mathematics)","wgRelevantArticleId":19447,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Group_(math)","wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":100000,"wgInternalRedirectTargetUrl":"/wiki/Group_(mathematics)","wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q83478","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGELevelingUpEnabledForUser":false}; RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","mediawiki.page.media","ext.scribunto.logs","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.21"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/1200px-Rubik%27s_cube.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1250"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/800px-Rubik%27s_cube.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="833"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/640px-Rubik%27s_cube.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="667"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Group (mathematics) - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Group_(mathematics)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Group_(mathematics)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Group_(mathematics)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Group_mathematics rootpage-Group_mathematics skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Main menu </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z">Main page</a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia">Contents</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events">Current events</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x">Random article</a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works">About Wikipedia</a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia">Contact us</a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia">Help</a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia">Learn to edit</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors">Community portal</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r">Recent changes</a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia">Upload file</a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages">Special pages</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"> Search </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Appearance </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class="">Donate</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Group+%28mathematics%29" title="You are encouraged to create an account and log in; however, it is not mandatory" class="">Create account</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Group+%28mathematics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class="">Log in</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Personal tools </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en">Donate</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Group+%28mathematics%29" title="You are encouraged to create an account and log in; however, it is not mandatory"> Create account</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Group+%28mathematics%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"> Log in</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing">learn more</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y">Contributions</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n">Talk</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Definition_and_illustration" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Definition_and_illustration"> <div class="vector-toc-text"> 1 Definition and illustration </div> </a> <button aria-controls="toc-Definition_and_illustration-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition and illustration subsection </button> <ul id="toc-Definition_and_illustration-sublist" class="vector-toc-list"> <li id="toc-First_example:_the_integers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#First_example:_the_integers"> <div class="vector-toc-text"> 1.1 First example: the integers </div> </a> <ul id="toc-First_example:_the_integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 1.2 Definition </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notation_and_terminology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notation_and_terminology"> <div class="vector-toc-text"> 1.3 Notation and terminology </div> </a> <ul id="toc-Notation_and_terminology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second_example:_a_symmetry_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second_example:_a_symmetry_group"> <div class="vector-toc-text"> 1.4 Second example: a symmetry group </div> </a> <ul id="toc-Second_example:_a_symmetry_group-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elementary_consequences_of_the_group_axioms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elementary_consequences_of_the_group_axioms"> <div class="vector-toc-text"> 3 Elementary consequences of the group axioms </div> </a> <button aria-controls="toc-Elementary_consequences_of_the_group_axioms-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Elementary consequences of the group axioms subsection </button> <ul id="toc-Elementary_consequences_of_the_group_axioms-sublist" class="vector-toc-list"> <li id="toc-Uniqueness_of_identity_element" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness_of_identity_element"> <div class="vector-toc-text"> 3.1 Uniqueness of identity element </div> </a> <ul id="toc-Uniqueness_of_identity_element-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniqueness_of_inverses" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniqueness_of_inverses"> <div class="vector-toc-text"> 3.2 Uniqueness of inverses </div> </a> <ul id="toc-Uniqueness_of_inverses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division"> <div class="vector-toc-text"> 3.3 Division </div> </a> <ul id="toc-Division-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Equivalent_definition_with_relaxed_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_definition_with_relaxed_axioms"> <div class="vector-toc-text"> 3.4 Equivalent definition with relaxed axioms </div> </a> <ul id="toc-Equivalent_definition_with_relaxed_axioms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_concepts"> <div class="vector-toc-text"> 4 Basic concepts </div> </a> <button aria-controls="toc-Basic_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Basic concepts subsection </button> <ul id="toc-Basic_concepts-sublist" class="vector-toc-list"> <li id="toc-Group_homomorphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_homomorphisms"> <div class="vector-toc-text"> 4.1 Group homomorphisms </div> </a> <ul id="toc-Group_homomorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subgroups"> <div class="vector-toc-text"> 4.2 Subgroups </div> </a> <ul id="toc-Subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cosets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cosets"> <div class="vector-toc-text"> 4.3 Cosets </div> </a> <ul id="toc-Cosets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quotient_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quotient_groups"> <div class="vector-toc-text"> 4.4 Quotient groups </div> </a> <ul id="toc-Quotient_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Presentations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Presentations"> <div class="vector-toc-text"> 4.5 Presentations </div> </a> <ul id="toc-Presentations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples_and_applications"> <div class="vector-toc-text"> 5 Examples and applications </div> </a> <button aria-controls="toc-Examples_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Examples and applications subsection </button> <ul id="toc-Examples_and_applications-sublist" class="vector-toc-list"> <li id="toc-Numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Numbers"> <div class="vector-toc-text"> 5.1 Numbers </div> </a> <ul id="toc-Numbers-sublist" class="vector-toc-list"> <li id="toc-Integers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Integers"> <div class="vector-toc-text"> 5.1.1 Integers </div> </a> <ul id="toc-Integers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rationals" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rationals"> <div class="vector-toc-text"> 5.1.2 Rationals </div> </a> <ul id="toc-Rationals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Modular_arithmetic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modular_arithmetic"> <div class="vector-toc-text"> 5.2 Modular arithmetic </div> </a> <ul id="toc-Modular_arithmetic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclic_groups"> <div class="vector-toc-text"> 5.3 Cyclic groups </div> </a> <ul id="toc-Cyclic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetry_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetry_groups"> <div class="vector-toc-text"> 5.4 Symmetry groups </div> </a> <ul id="toc-Symmetry_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_linear_group_and_representation_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_linear_group_and_representation_theory"> <div class="vector-toc-text"> 5.5 General linear group and representation theory </div> </a> <ul id="toc-General_linear_group_and_representation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Galois_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Galois_groups"> <div class="vector-toc-text"> 5.6 Galois groups </div> </a> <ul id="toc-Galois_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Finite_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Finite_groups"> <div class="vector-toc-text"> 6 Finite groups </div> </a> <button aria-controls="toc-Finite_groups-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Finite groups subsection </button> <ul id="toc-Finite_groups-sublist" class="vector-toc-list"> <li id="toc-Finite_abelian_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_abelian_groups"> <div class="vector-toc-text"> 6.1 Finite abelian groups </div> </a> <ul id="toc-Finite_abelian_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_groups"> <div class="vector-toc-text"> 6.2 Simple groups </div> </a> <ul id="toc-Simple_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_of_finite_simple_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification_of_finite_simple_groups"> <div class="vector-toc-text"> 6.3 Classification of finite simple groups </div> </a> <ul id="toc-Classification_of_finite_simple_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Groups_with_additional_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Groups_with_additional_structure"> <div class="vector-toc-text"> 7 Groups with additional structure </div> </a> <button aria-controls="toc-Groups_with_additional_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Groups with additional structure subsection </button> <ul id="toc-Groups_with_additional_structure-sublist" class="vector-toc-list"> <li id="toc-Topological_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_groups"> <div class="vector-toc-text"> 7.1 Topological groups </div> </a> <ul id="toc-Topological_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_groups"> <div class="vector-toc-text"> 7.2 Lie groups </div> </a> <ul id="toc-Lie_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 8 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 10 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 11 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-General_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> 12.1 General references </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_references"> <div class="vector-toc-text"> 12.2 Special references </div> </a> <ul id="toc-Special_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_references"> <div class="vector-toc-text"> 12.3 Historical references </div> </a> <ul id="toc-Historical_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Group (mathematics)</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 84 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-84" aria-hidden="true" > 84 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Groep_(wiskunde)" title="Groep (wiskunde) – Afrikaans" lang="af" hreflang="af" data-title="Groep (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="زمرة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="زمرة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A7%81%E0%A6%AA_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="গ্রুপ (গণিত) – Bangla" lang="bn" hreflang="bn" data-title="গ্রুপ (গণিত)" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C3%BBn_(s%C3%B2%CD%98-ha%CC%8Dk)" title="Kûn (sò͘-ha̍k) – Minnan" lang="nan" hreflang="nan" data-title="Kûn (sò͘-ha̍k)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D3%A9%D1%80%D0%BA%D3%A9%D0%BC_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Төркөм (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Төркөм (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Група (алгебра) – Belarusian" lang="be" hreflang="be" data-title="Група (алгебра)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Група (алгебра) – Bulgarian" lang="bg" hreflang="bg" data-title="Група (алгебра)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_(matem%C3%A0tiques)" title="Grup (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Grup (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D1%88%D0%BA%C4%83%D0%BD_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ушкăн (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Ушкăн (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://cs.wikipedia.org/wiki/Grupa" title="Grupa – Czech" lang="cs" hreflang="cs" data-title="Grupa" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gr%C5%B5p_(mathemateg)" title="Grŵp (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Grŵp (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gruppe_(matematik)" title="Gruppe (matematik) – Danish" lang="da" hreflang="da" data-title="Gruppe (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gruppe_(Mathematik)" title="Gruppe (Mathematik) – German" lang="de" hreflang="de" data-title="Gruppe (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/R%C3%BChm_(matemaatika)" title="Rühm (matemaatika) – Estonian" lang="et" hreflang="et" data-title="Rühm (matemaatika)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BC%CE%AC%CE%B4%CE%B1" title="Ομάδα – Greek" lang="el" hreflang="el" data-title="Ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_(matem%C3%A1tica)" title="Grupo (matemática) – Spanish" lang="es" hreflang="es" data-title="Grupo (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grupo_(algebro)" title="Grupo (algebro) – Esperanto" lang="eo" hreflang="eo" data-title="Grupo (algebro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde_(matematika)" title="Talde (matematika) – Basque" lang="eu" hreflang="eu" data-title="Talde (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="گروه (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="گروه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques) – French" lang="fr" hreflang="fr" data-title="Groupe (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Gr%C3%BApa_(matamaitic)" title="Grúpa (matamaitic) – Irish" lang="ga" hreflang="ga" data-title="Grúpa (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Grupo_(matem%C3%A1ticas)" title="Grupo (matemáticas) – Galician" lang="gl" hreflang="gl" data-title="Grupo (matemáticas)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0_(%EC%88%98%ED%95%99)" title="군 (수학) – Korean" lang="ko" hreflang="ko" data-title="군 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BD%D5%B8%D6%82%D5%B4%D5%A2_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Խումբ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Խումբ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%82%E0%A4%B9_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="समूह (गणितशास्त्र) – Hindi" lang="hi" hreflang="hi" data-title="समूह (गणितशास्त्र)" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) – Croatian" lang="hr" hreflang="hr" data-title="Grupa (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_(matematika)" title="Grup (matematika) – Indonesian" lang="id" hreflang="id" data-title="Grup (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_(mathematica)" title="Gruppo (mathematica) – Interlingua" lang="ia" hreflang="ia" data-title="Gruppo (mathematica)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Gr%C3%BApa" title="Grúpa – Icelandic" lang="is" hreflang="is" data-title="Grúpa" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_(matematica)" title="Gruppo (matematica) – Italian" lang="it" hreflang="it" data-title="Gruppo (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="חבורה (מבנה אלגברי) – Hebrew" lang="he" hreflang="he" data-title="חבורה (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%8D%E0%B2%B0%E0%B3%82%E0%B2%AA%E0%B3%8D" title="ಗ್ರೂಪ್ – Kannada" lang="kn" hreflang="kn" data-title="ಗ್ರೂಪ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AF%E1%83%92%E1%83%A3%E1%83%A4%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ჯგუფი (მათემატიკა) – Georgian" lang="ka" hreflang="ka" data-title="ჯგუფი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Топ (математика) – Kazakh" lang="kk" hreflang="kk" data-title="Топ (математика)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Caterva_(mathematica)" title="Caterva (mathematica) – Latin" lang="la" hreflang="la" data-title="Caterva (mathematica)" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Grupa_(matem%C4%81tika)" title="Grupa (matemātika) – Latvian" lang="lv" hreflang="lv" data-title="Grupa (matemātika)" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Grupp_(Algeber)" title="Grupp (Algeber) – Luxembourgish" lang="lb" hreflang="lb" data-title="Grupp (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Grup%C4%97_(algebra)" title="Grupė (algebra) – Lithuanian" lang="lt" hreflang="lt" data-title="Grupė (algebra)" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Grupp_(matem%C3%A0tica)" title="Grupp (matemàtica) – Lombard" lang="lmo" hreflang="lmo" data-title="Grupp (matemàtica)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Csoport_(matematika)" title="Csoport (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Csoport (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Vory_(matematika)" title="Vory (matematika) – Malagasy" lang="mg" hreflang="mg" data-title="Vory (matematika)" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target">Malagasy</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D" title="ഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Grupp_(matematika)" title="Grupp (matematika) – Maltese" lang="mt" hreflang="mt" data-title="Grupp (matematika)" data-language-autonym="Malti" data-language-local-name="Maltese" class="interlanguage-link-target">Malti</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kumpulan_(matematik)" title="Kumpulan (matematik) – Malay" lang="ms" hreflang="ms" data-title="Kumpulan (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%91%D2%AF%D0%BB%D1%8D%D0%B3_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA)" title="Бүлэг (математик) – Mongolian" lang="mn" hreflang="mn" data-title="Бүлэг (математик)" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Groep_(wiskunde)" title="Groep (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Groep (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%BE%A4_(%E6%95%B0%E5%AD%A6)" title="群 (数学) – Japanese" lang="ja" hreflang="ja" data-title="群 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Sk%C3%B6%C3%B6l_(Matematiik)" title="Skööl (Matematiik) – Northern Frisian" lang="frr" hreflang="frr" data-title="Skööl (Matematiik)" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gruppe_(matematikk)" title="Gruppe (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gruppe (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_gruppe" title="Matematisk gruppe – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk gruppe" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Grupe_(matematike)" title="Grupe (matematike) – Novial" lang="nov" hreflang="nov" data-title="Grupe (matematike)" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Grop_(matematicas)" title="Grop (matematicas) – Occitan" lang="oc" hreflang="oc" data-title="Grop (matematicas)" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="گروہ (ریاضی) – Western Punjabi" lang="pnb" hreflang="pnb" data-title="گروہ (ریاضی)" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Strop" title="Strop – Piedmontese" lang="pms" hreflang="pms" data-title="Strop" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_(matematyka)" title="Grupa (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Grupa (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_(matem%C3%A1tica)" title="Grupo (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Grupo (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ro.wikipedia.org/wiki/Grup_(matematic%C4%83)" title="Grup (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Grup (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Группа (математика) – Russian" lang="ru" hreflang="ru" data-title="Группа (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Gruppu_(matimatica)" title="Gruppu (matimatica) – Sicilian" lang="scn" hreflang="scn" data-title="Gruppu (matimatica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Group_(mathematics)" title="Group (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Group (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) – Slovak" lang="sk" hreflang="sk" data-title="Grupa (matematika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Grupa" title="Grupa – Slovenian" lang="sl" hreflang="sl" data-title="Grupa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Grupa_(matymatyka)" title="Grupa (matymatyka) – Silesian" lang="szl" hreflang="szl" data-title="Grupa (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target">Ślůnski</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%88%D9%BE_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="گرووپ (ماتماتیک) – Central Kurdish" lang="ckb" hreflang="ckb" data-title="گرووپ (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Група (математика) – Serbian" lang="sr" hreflang="sr" data-title="Група (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Grupa (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ryhm%C3%A4_(algebra)" title="Ryhmä (algebra) – Finnish" lang="fi" hreflang="fi" data-title="Ryhmä (algebra)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Grupp_(matematik)" title="Grupp (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Grupp (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Grupo_(matematika)" title="Grupo (matematika) – Tagalog" lang="tl" hreflang="tl" data-title="Grupo (matematika)" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="குலம் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="குலம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tagrumma_(tusnakt)" title="Tagrumma (tusnakt) – Kabyle" lang="kab" hreflang="kab" data-title="Tagrumma (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="Kabyle" class="interlanguage-link-target">Taqbaylit</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B8%E0%B8%9B_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="กรุป (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="กรุป (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D3%AF%D2%B3_(%D1%80%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82)" title="Гурӯҳ (риёзиёт) – Tajik" lang="tg" hreflang="tg" data-title="Гурӯҳ (риёзиёт)" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target">Тоҷикӣ</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Grup_(matematik)" title="Grup (matematik) – Turkish" lang="tr" hreflang="tr" data-title="Grup (matematik)" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Група (математика) – Ukrainian" lang="uk" hreflang="uk" data-title="Група (математика)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="گروہ (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="گروہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_(to%C3%A1n_h%E1%BB%8Dc)" title="Nhóm (toán học) – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-vo mw-list-item"><a href="https://vo.wikipedia.org/wiki/Grup" title="Grup – Volapük" lang="vo" hreflang="vo" data-title="Grup" data-language-autonym="Volapük" data-language-local-name="Volapük" class="interlanguage-link-target">Volapük</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%BE%A4_(%E4%BB%A3%E6%95%B8)" title="群 (代數) – Literary Chinese" lang="lzh" hreflang="lzh" data-title="群 (代數)" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Groep_(algebra)" title="Groep (algebra) – West Flemish" lang="vls" hreflang="vls" data-title="Groep (algebra)" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target">West-Vlams</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BE%A4" title="群 – Wu" lang="wuu" hreflang="wuu" data-title="群" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%A8%D7%95%D7%A4%D7%A2_(%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7)" title="גרופע (מאטעמאטיק) – Yiddish" lang="yi" hreflang="yi" data-title="גרופע (מאטעמאטיק)" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target">ייִדיש</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%BE%A3" title="羣 – Cantonese" lang="yue" hreflang="yue" data-title="羣" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BE%A4" title="群 – Chinese" lang="zh" hreflang="zh" data-title="群" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q83478#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Group_(mathematics)" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Group_(mathematics)" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t">Talk</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >English </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Group_(mathematics)">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&action=history" title="Past revisions of this page [h]" accesskey="h">View history</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Tools </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Group_(mathematics)">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&action=edit" title="Edit this page [e]" accesskey="e">Edit</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&action=history">View history</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Group_(mathematics)" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j">What links here</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Group_(mathematics)" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k">Related changes</a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&oldid=1280089198" title="Permanent link to this revision of this page">Permanent link</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&action=info" title="More information about this page">Page information</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Group_%28mathematics%29&id=1280089198&wpFormIdentifier=titleform" title="Information on how to cite this page">Cite this page</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGroup_%28mathematics%29">Get shortened URL</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FGroup_%28mathematics%29">Download QR code</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Group_%28mathematics%29&action=show-download-screen" title="Download this page as a PDF file">Download as PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Group_(mathematics)&printable=yes" title="Printable version of this page [p]" accesskey="p">Printable version</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Group_theory" hreflang="en">Wikimedia Commons</a></li><li class="wb-otherproject-link wb-otherproject-wikibooks mw-list-item"><a href="https://en.wikibooks.org/wiki/Abstract_Algebra/Group_Theory/Group" hreflang="en">Wikibooks</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q83478" title="Structured data on this page hosted by Wikidata [g]" accesskey="g">Wikidata item</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-featured-star" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Featured_articles*" title="This is a featured article. Click here for more information."><img alt="Featured article" src="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/20px-Cscr-featured.svg.png" decoding="async" width="20" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/30px-Cscr-featured.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/e/e7/Cscr-featured.svg/40px-Cscr-featured.svg.png 2x" data-file-width="466" data-file-height="443" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle">(Redirected from <a href="/w/index.php?title=Group_(math)&redirect=no" class="mw-redirect" title="Group (math)">Group (math)</a>)</div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Set with associative invertible operation</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about basic notions of groups in mathematics. For a more advanced treatment, see <a href="/wiki/Group_theory" title="Group theory">Group theory</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Rubik%27s_cube.svg" class="mw-file-description"><img alt="A Rubik's cube with one side rotated" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/220px-Rubik%27s_cube.svg.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/330px-Rubik%27s_cube.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/440px-Rubik%27s_cube.svg.png 2x" data-file-width="480" data-file-height="500" /></a><figcaption>The manipulations of the <a href="/wiki/Rubik%27s_Cube" title="Rubik's Cube">Rubik's Cube</a> form the <a href="/wiki/Rubik%27s_Cube_group" title="Rubik's Cube group">Rubik's Cube group</a>.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a group is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> with a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> that satisfies the following constraints: the operation is <a href="/wiki/Associative_property" title="Associative property">associative</a>, it has an <a href="/wiki/Identity_element" title="Identity element">identity element</a>, and every element of the set has an <a href="/wiki/Inverse_element" title="Inverse element">inverse element</a>. Many <a href="/wiki/Mathematical_structure" title="Mathematical structure">mathematical structures</a> are groups endowed with other properties. For example, the <a href="/wiki/Integer" title="Integer">integers</a> with the <a href="/wiki/Addition" title="Addition">addition operation</a> form an <a href="/wiki/Infinite_set" title="Infinite set">infinite</a> group, which is <a href="/wiki/Generating_set" class="mw-redirect" title="Generating set">generated by a single element</a> called ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠ (these properties characterize the integers in a unique way). The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, <a href="/wiki/Geometric_shape" class="mw-redirect" title="Geometric shape">geometric shapes</a> and <a href="/wiki/Polynomial_root" class="mw-redirect" title="Polynomial root">polynomial roots</a>. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.<a href="#cite_note-FOOTNOTEHerstein197526§2-1">[1]</a><a href="#cite_note-FOOTNOTEHall19671§1.1:_"The_idea_of_a_group_is_one_which_pervades_the_whole_of_mathematics_both_[[pure_mathematics|pure]]_and_[[applied_mathematics|applied]]."-2">[2]</a> In <a href="/wiki/Geometry" title="Geometry">geometry</a>, groups arise naturally in the study of <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a> and <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformations</a>: The symmetries of an object form a group, called the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of the object, and the transformations of a given type form a general group. <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> appear in symmetry groups in geometry, and also in the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> of <a href="/wiki/Particle_physics" title="Particle physics">particle physics</a>. The <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a> is a Lie group consisting of the symmetries of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>. <a href="/wiki/Point_group" title="Point group">Point groups</a> describe <a href="/wiki/Molecular_symmetry" title="Molecular symmetry">symmetry in molecular chemistry</a>. The concept of a group arose in the study of <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a>, starting with <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> in the 1830s, who introduced the term group (French: groupe) for the symmetry group of the <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots</a> of an equation, now called a <a href="/wiki/Galois_group" title="Galois group">Galois group</a>. After contributions from other fields such as <a href="/wiki/Number_theory" title="Number theory">number theory</a> and geometry, the group notion was generalized and firmly established around 1870. Modern <a href="/wiki/Group_theory" title="Group theory">group theory</a>—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>, <a href="/wiki/Quotient_group" title="Quotient group">quotient groups</a> and <a href="/wiki/Simple_group" title="Simple group">simple groups</a>. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view of <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> (that is, through the <a href="/wiki/Group_representation" title="Group representation">representations of the group</a>) and of <a href="/wiki/Computational_group_theory" title="Computational group theory">computational group theory</a>. A theory has been developed for <a href="/wiki/Finite_group" title="Finite group">finite groups</a>, which culminated with the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>, completed in 2004. Since the mid-1980s, <a href="/wiki/Geometric_group_theory" title="Geometric group theory">geometric group theory</a>, which studies <a href="/wiki/Finitely_generated_group" title="Finitely generated group">finitely generated groups</a> as geometric objects, has become an active area in group theory. <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1246091330">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><table class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Group theory <a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Zn</li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> Sn</li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> An</li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> Dn</li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group">p-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />)</li><li>SL(2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(n)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(n)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(n)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(n)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(n)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(n)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(n)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(n)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E6</a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E7</a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E8</a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="display:block;margin-bottom:0.35em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Group</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a class="mw-selflink selflink">Group</a></li> <li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> / <a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Racks_and_quandles" title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <a href="/wiki/Group_theory" title="Group theory">Group theory</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div> <ul><li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Definition_and_illustration">Definition and illustration</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=1" title="Edit section: Definition and illustration">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="First_example:_the_integers">First example: the integers</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=2" title="Edit section: First example: the integers">edit</a>]</div> One of the more familiar groups is the set of <a href="/wiki/Integer" title="Integer">integers</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713cb71667a2a73d3ed32c43fbae0b991a1be2d2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.841ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}" /> together with <a href="/wiki/Addition" title="Addition">addition</a>.<a href="#cite_note-FOOTNOTELang2005360App._2-3">[3]</a> For any two integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" />⁠, the <a href="/wiki/Summation" title="Summation">sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" /> is also an integer; this <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closure</a> property says that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /> is a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />⁠. The following properties of integer addition serve as a model for the group axioms in the definition below. <ul><li>For all integers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />⁠, one has ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)+c=a+(b+c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)+c=a+(b+c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46b7b8d31d5845966e6abdbb030c73f343c17d4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.547ex; height:2.843ex;" alt="{\displaystyle (a+b)+c=a+(b+c)}" />⁠. Expressed in words, adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> first, and then adding the result to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /> gives the same final result as adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> to the sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />⁠. This property is known as <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a>.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> is any integer, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0+a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0+a=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba565116cf2f2d984f7b8365b054b70eb8f89308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle 0+a=a}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+0=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+0=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4564e28f0f8274644ca4e58664c0593ed48de541" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.561ex; height:2.343ex;" alt="{\displaystyle a+0=a}" />⁠. <a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a> is called the <a href="/wiki/Identity_element" title="Identity element">identity element</a> of addition because adding it to any integer returns the same integer.</li> <li>For every integer ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, there is an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5fde0c70831149e78bf64025ae44888a5b251da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.329ex; height:2.343ex;" alt="{\displaystyle a+b=0}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b+a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+a=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/429db75a431dd53fb698253d74eab6ab4e6e6bc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.329ex; height:2.343ex;" alt="{\displaystyle b+a=0}" />⁠. The integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> is called the <a href="/wiki/Inverse_element" title="Inverse element">inverse element</a> of the integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and is denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}" />⁠.</li></ul> The integers, together with the operation ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" />⁠, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed. <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=3" title="Edit section: Definition">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1224211176">.mw-parser-output .quotebox{background-color:#F9F9F9;border:1px solid #aaa;box-sizing:border-box;padding:10px;font-size:88%;max-width:100%}.mw-parser-output .quotebox.floatleft{margin:.5em 1.4em .8em 0}.mw-parser-output .quotebox.floatright{margin:.5em 0 .8em 1.4em}.mw-parser-output .quotebox.centered{overflow:hidden;position:relative;margin:.5em auto .8em auto}.mw-parser-output .quotebox.floatleft span,.mw-parser-output .quotebox.floatright span{font-style:inherit}.mw-parser-output .quotebox>blockquote{margin:0;padding:0;border-left:0;font-family:inherit;font-size:inherit}.mw-parser-output .quotebox-title{text-align:center;font-size:110%;font-weight:bold}.mw-parser-output .quotebox-quote>:first-child{margin-top:0}.mw-parser-output .quotebox-quote:last-child>:last-child{margin-bottom:0}.mw-parser-output .quotebox-quote.quoted:before{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" “ ";vertical-align:-45%;line-height:0}.mw-parser-output .quotebox-quote.quoted:after{font-family:"Times New Roman",serif;font-weight:bold;font-size:large;color:gray;content:" ” ";line-height:0}.mw-parser-output .quotebox .left-aligned{text-align:left}.mw-parser-output .quotebox .right-aligned{text-align:right}.mw-parser-output .quotebox .center-aligned{text-align:center}.mw-parser-output .quotebox .quote-title,.mw-parser-output .quotebox .quotebox-quote{display:block}.mw-parser-output .quotebox cite{display:block;font-style:normal}@media screen and (max-width:640px){.mw-parser-output .quotebox{width:100%!important;margin:0 0 .8em!important;float:none!important}}</style><div class="quotebox pullquote floatright" style="width:33%; ;"> <blockquote class="quotebox-quote left-aligned" style=""> The axioms for a group are short and natural ... Yet somehow hidden behind these axioms is the <a href="/wiki/Monster_group" title="Monster group">monster simple group</a>, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists. </blockquote> <div style="padding-bottom: 0; padding-top: 0.5em"><cite class="left-aligned" style=""><a href="/wiki/Richard_Borcherds" title="Richard Borcherds">Richard Borcherds</a>, Mathematicians: An Outer View of the Inner World<a href="#cite_note-FOOTNOTECook200924-4">[4]</a></cite></div> </div> A group is a non-empty <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> together with a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, here denoted "⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" />⁠", that combines any two <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> to form an element of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620419d3ed53abc98659a5fc0f3a5eb6177830ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}" />⁠, such that the following three requirements, known as group axioms, are satisfied:<a href="#cite_note-FOOTNOTEArtin201840§2.2-5">[5]</a><a href="#cite_note-FOOTNOTELang2002p._3,_I.§1_and_p._7,_I.§2-6">[6]</a><a href="#cite_note-FOOTNOTELang200516II.§1-7">[7]</a><a href="#cite_note-8">[a]</a> <dl><dt>Associativity</dt> <dd>For all ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />⁠ in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, one has ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e48cd711b9a1eb1c3a2372ba01fa48ca7e262a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.902ex; height:2.843ex;" alt="{\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}" />⁠.</dd> <dt>Identity element</dt> <dd>There exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> such that, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, one has ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot a=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a692f53d86977a514677a70e5f4dec9ae7c9ebd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.321ex; height:1.676ex;" alt="{\displaystyle e\cdot a=a}" />⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot e=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>e</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot e=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efda0c21947d4ebdb68a0e76149c32a53ee357dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.321ex; height:1.676ex;" alt="{\displaystyle a\cdot e=a}" />⁠.</dd> <dd>Such an element is unique (<a href="#Uniqueness_of_identity_element">see below</a>). It is called the identity element (or sometimes neutral element) of the group.</dd> <dt>Inverse element</dt> <dd>For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b18f2558746cf54b8b81b4abae79d02bb526ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.088ex; height:2.176ex;" alt="{\displaystyle a\cdot b=e}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\cdot a=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\cdot a=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3df9f7f39c0f0d626391d147dedc7bd55f0720b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.088ex; height:2.176ex;" alt="{\displaystyle b\cdot a=e}" />⁠, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /> is the identity element.</dd> <dd>For each ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> is unique (<a href="#Uniqueness_of_inverses">see below</a>); it is called the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and is commonly denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5709c8d86f7fec8fb86069bf5d15a9eabe564e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-1}}" />⁠.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Notation_and_terminology">Notation and terminology</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=4" title="Edit section: Notation and terminology">edit</a>]</div> Formally, a group is an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> of a set and a binary operation on this set that satisfies the <a href="/wiki/Group_axioms" class="mw-redirect" title="Group axioms">group axioms</a>. The set is called the underlying set of the group, and the operation is called the group operation or the group law. A group and its underlying set are thus two different <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>. To avoid cumbersome notation, it is common to <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse notation</a> by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation. For example, consider the set of <a href="/wiki/Real_number" title="Real number">real numbers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />⁠, which has the operations of addition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}" /> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}" />⁠. Formally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is a set, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b33b2c9358cbd7bad20aa0b18651d3bba582c09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+)}" /> is a group, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,+,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,+,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed3304e521dac824c9b33476be16be5fadde282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.01ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,+,\cdot )}" /> is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. But it is common to write <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> to denote any of these three objects. The additive group of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is the group whose underlying set is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> and whose operation is addition. The multiplicative group of the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" /> is the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0a3db7fcd70511998574e8060145d0723e458b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.189ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{\times }}" /> whose underlying set is the set of nonzero real numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \smallsetminus \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \smallsetminus \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ef60217cde8399eb7841dae03763d676e6e2a55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.006ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} \smallsetminus \{0\}}" /> and whose operation is multiplication. More generally, one speaks of an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" />⁠, and the inverse of an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> is denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae55e66aeffc525917eed885b4b753ba5a7f8b3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle -x}" />⁠. Similarly, one speaks of a multiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠, and the inverse of an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> is denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf91609f1a0b7847e108023b015cb6b0d567821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle x^{-1}}" />⁠. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}" /> instead of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620419d3ed53abc98659a5fc0f3a5eb6177830ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}" />⁠. The definition of a group does not require that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=b\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=b\cdot a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b7dede7493e0231b3ad6ff9b54f4eae954108" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.911ex; height:2.176ex;" alt="{\displaystyle a\cdot b=b\cdot a}" /> for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠. If this additional condition holds, then the operation is said to be <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, and the group is called an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used. Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, the operation is often <a href="/wiki/Function_composition" title="Function composition">function composition</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∘</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\circ g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2f61ca7838709fbae07dce9c0d513770f10cfae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\circ g}" />⁠; then the identity may be denoted id. In the more specific cases of <a href="/wiki/Geometric_transformation" title="Geometric transformation">geometric transformation</a> groups, <a href="/wiki/Symmetry_(mathematics)" class="mw-redirect" title="Symmetry (mathematics)">symmetry</a> groups, <a href="/wiki/Permutation_group" title="Permutation group">permutation groups</a>, and <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism groups</a>, the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" /> is often omitted, as for multiplicative groups. Many other variants of notation may be encountered. <div class="mw-heading mw-heading3"><h3 id="Second_example:_a_symmetry_group">Second example: a symmetry group</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=5" title="Edit section: Second example: a symmetry group">edit</a>]</div> Two figures in the <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> are <a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">congruent</a> if one can be changed into the other using a combination of <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a>, <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a>, and <a href="/wiki/Translation_(geometry)" title="Translation (geometry)">translations</a>. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called <a href="/wiki/Symmetry" title="Symmetry">symmetries</a>. A <a href="/wiki/Square" title="Square">square</a> has eight symmetries. These are: <table class="wikitable" style="text-align:center;"> <caption>The elements of the symmetry group of the square, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠. Vertices are identified by color or number. </caption> <tbody><tr> <td><a href="/wiki/File:Group_D8_id.svg" class="mw-file-description"><img alt="A square with its four corners marked by 1 to 4" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Group_D8_id.svg/140px-Group_D8_id.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Group_D8_id.svg/210px-Group_D8_id.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Group_D8_id.svg/280px-Group_D8_id.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> (keeping it as it is)</td> <td><a href="/wiki/File:Group_D8_90.svg" class="mw-file-description"><img alt="The square is rotated by 90° clockwise; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Group_D8_90.svg/140px-Group_D8_90.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Group_D8_90.svg/210px-Group_D8_90.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Group_D8_90.svg/280px-Group_D8_90.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> (rotation by 90° clockwise)</td> <td><a href="/wiki/File:Group_D8_180.svg" class="mw-file-description"><img alt="The square is rotated by 180° clockwise; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Group_D8_180.svg/140px-Group_D8_180.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Group_D8_180.svg/210px-Group_D8_180.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Group_D8_180.svg/280px-Group_D8_180.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> (rotation by 180°)</td> <td><a href="/wiki/File:Group_D8_270.svg" class="mw-file-description"><img alt="The square is rotated by 270° clockwise; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Group_D8_270.svg/140px-Group_D8_270.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Group_D8_270.svg/210px-Group_D8_270.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Group_D8_270.svg/280px-Group_D8_270.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> (rotation by 270° clockwise) </td></tr> <tr> <td><a href="/wiki/File:Group_D8_fv.svg" class="mw-file-description"><img alt="The square is reflected vertically; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Group_D8_fv.svg/140px-Group_D8_fv.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Group_D8_fv.svg/210px-Group_D8_fv.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Group_D8_fv.svg/280px-Group_D8_fv.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> (vertical reflection)</td> <td> <a href="/wiki/File:Group_D8_fh.svg" class="mw-file-description"><img alt="The square is reflected horizontally; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Group_D8_fh.svg/140px-Group_D8_fh.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Group_D8_fh.svg/210px-Group_D8_fh.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Group_D8_fh.svg/280px-Group_D8_fh.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /> (horizontal reflection) </td> <td> <a href="/wiki/File:Group_D8_f13.svg" class="mw-file-description"><img alt="The square is reflected along the SW–NE diagonal; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Group_D8_f13.svg/140px-Group_D8_f13.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Group_D8_f13.svg/210px-Group_D8_f13.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Group_D8_f13.svg/280px-Group_D8_f13.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /> (diagonal reflection) </td> <td> <a href="/wiki/File:Group_D8_f24.svg" class="mw-file-description"><img alt="The square is reflected along the SE–NW diagonal; the corners are enumerated accordingly." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Group_D8_f24.svg/140px-Group_D8_f24.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Group_D8_f24.svg/210px-Group_D8_f24.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Group_D8_f24.svg/280px-Group_D8_f24.svg.png 2x" data-file-width="160" data-file-height="135" /></a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /> (counter-diagonal reflection) </td></tr></tbody></table> <ul><li>the <a href="/wiki/Identity_operation" class="mw-redirect" title="Identity operation">identity operation</a> leaving everything unchanged, denoted id;</li> <li>rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" />⁠, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" />⁠, respectively;</li> <li>reflections about the horizontal and vertical middle line (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" />⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" />⁠), or through the two <a href="/wiki/Diagonal" title="Diagonal">diagonals</a> (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" />⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" />⁠).</li></ul> <div style="clear:both;" class=""></div> These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> sends a point to its rotation 90° clockwise around the square's center, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /> sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> of degree four, denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠. The underlying set of the group is the above set of symmetries, and the group operation is function composition.<a href="#cite_note-FOOTNOTEHerstein197554§2.6-9">[8]</a> Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> is written symbolically from right to left as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\circ a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∘</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\circ a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12766887e6a74b8b91e2df5be1132f75df723bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.422ex; height:2.176ex;" alt="{\displaystyle b\circ a}" /> ("apply the symmetry <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> after performing the symmetry ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠"). This is the usual notation for composition of functions. A <a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a> lists the results of all such compositions possible. For example, rotating by 270° clockwise (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" />⁠) and then reflecting horizontally (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" />⁠) is the same as performing a reflection along the diagonal (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" />⁠). Using the above symbols, highlighted in blue in the Cayley table: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b4133c453ae73e1cad2013612edf9d61c28e117" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.614ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }\circ r_{3}=f_{\mathrm {d} }.}" /> <table class="wikitable" style="float:right; text-align:center; margin:.5em 0 .5em 1em; width:40ex; height:40ex;"> <caption><a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> </caption> <tbody><tr> <th style="width:12%; background:#fdd; border-top:solid black 2px; border-left:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \circ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∘</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \circ }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99add39d2b681e2de7ff62422c32704a05c7ec31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.125ex; margin-bottom: -0.297ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle \circ }" /> </th> <th style="background:#fdd; border-top:solid black 2px; width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> </th> <th style="background:#fdd; border-top:solid black 2px; width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> </th> <th style="background:#fdd; border-top:solid black 2px; width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> </th> <th style="background:#fdd; border-right:solid black 2px; border-top:solid black 2px; width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> </th> <th style="width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /></th> <th style="width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></th> <th style="width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /></th> <th style="width:11%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /> </th></tr> <tr> <th style="background:#FDD; border-left:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> </th> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> </td> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> </td> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> </td> <td style="background:#FDD; border-right:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /> </td> <td style="background:#FFFC93; border-right:solid black 2px; border-left:solid black 2px; border-top:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /> </td></tr> <tr> <th style="background:#FDD; border-left:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> </th> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> </td> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> </td> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> </td> <td style="background:#FDD; border-right:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> </td> <td style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /> </td></tr> <tr style="height:10%"> <th style="background:#FDD; border-left:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> </th> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> </td> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> </td> <td style="background:#FDD;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> </td> <td style="background:#FDD; border-right:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /> </td> <td style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /> </td></tr> <tr style="height:10%"> <th style="background:#FDD; border-bottom:solid black 2px; border-left:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> </th> <td style="background:#FDD; border-bottom:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> </td> <td style="background:#FDD; border-bottom:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> </td> <td style="background:#FDD; border-bottom:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> </td> <td style="background:#FDD; border-right:solid black 2px; border-bottom:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /> </td> <td style="background:#FFFC93; border-right:solid black 2px; border-left:solid black 2px; border-bottom:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> </td></tr> <tr style="height:10%"> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> </td></tr> <tr style="height:10%"> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> </td> <td style="background:#BACDFF; border: solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> </td></tr> <tr style="height:10%"> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> </td></tr> <tr style="height:10%"> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /> </th> <td style="background:#9DFF93; border-left: solid black 2px; border-bottom: solid black 2px; border-top: solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" /> </td> <td style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> </td> <td style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" /> </td> <td style="background:#9DFF93; border-bottom:solid black 2px; border-top:solid black 2px; border-right:solid black 2px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" /> </td></tr> <tr> <td colspan="9" style="text-align:left">The elements ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" />⁠, and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" />⁠ form a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> whose Cayley table is highlighted in <style data-mw-deduplicate="TemplateStyles:r981673959">.mw-parser-output .legend{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .legend-color{display:inline-block;min-width:1.25em;height:1.25em;line-height:1.25;margin:1px 0;text-align:center;border:1px solid black;background-color:transparent;color:black}.mw-parser-output .legend-text{}</style>  red (upper left region). A left and right <a href="/wiki/Coset" title="Coset">coset</a> of this subgroup are highlighted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" />  green (in the last row) and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" />  yellow (last column), respectively. The result of the composition ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }\circ r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }\circ r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a0532d9523f5bea99304d1846d95034eaefcfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.583ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }\circ r_{3}}" />⁠, the symmetry ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" />⁠, is highlighted in <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r981673959" />  blue (below table center). </td></tr></tbody></table> Given this set of symmetries and the described operation, the group axioms can be understood as follows. Binary operation: Composition is a binary operation. That is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\circ b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∘</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\circ b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f270b24a930a6546b42f355ad905d2d7a26d4b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.422ex; height:2.176ex;" alt="{\displaystyle a\circ b}" /> is a symmetry for any two symmetries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" />⁠. For example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5055010135999617ee4029dd339e5088cbb16304" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.43ex; height:2.509ex;" alt="{\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} },}" /> that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" />⁠). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table. Associativity: The associativity axiom deals with composing more than two symmetries: Starting with three elements ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" />⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />⁠ of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> into a single symmetry, then to compose that symmetry with ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />⁠. The other way is to first compose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" />⁠, then to compose the resulting symmetry with ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠. These two ways must give always the same result, that is, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∘</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>∘</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∘</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53d7529dd3ca7b4dbf22e04404133b6d5a5353d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.611ex; height:2.843ex;" alt="{\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}" /> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>∘</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eafed02ad895bd583f0019c45cdab3cc56ac696d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.751ex; height:2.843ex;" alt="{\displaystyle (f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}=f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})}" /> can be checked using the Cayley table: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>∘</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42ff84f054eefa3385577772bde2749ca4fe12a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.29ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}" /> Identity element: The identity element is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" />⁠, as it does not change any symmetry <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> when composed with it either on the left or on the right. Inverse element: Each symmetry has an inverse: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" />⁠, the reflections ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9c0930663d86c724b4b4df4dc319b824c5586f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d335d9b5c1ed4eabacab6fb6eded88e6b9d115dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {d} }}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654588e10fc524524c7ab9f979a0ece43bb96def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.101ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }}" />⁠ and the 180° rotation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> are their own inverse, because performing them twice brings the square back to its original orientation. The rotations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table. In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠, as, for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>∘</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/918bf1cbb26fa603ec3d00c1109f62dd36486b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.783ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }}" /> but ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∘</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92f014518367b5098e7007703b9e5e6f57e71d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.967ex; height:2.509ex;" alt="{\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }}" />⁠. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> is not abelian. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=6" title="Edit section: History">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_group_theory" title="History of group theory">History of group theory</a></div> The modern concept of an <a href="/wiki/Abstract_group" class="mw-redirect" title="Abstract group">abstract group</a> developed out of several fields of mathematics.<a href="#cite_note-FOOTNOTEWussing2007-10">[9]</a><a href="#cite_note-FOOTNOTEKleiner1986-11">[10]</a><a href="#cite_note-FOOTNOTESmith1906-12">[11]</a> The original motivation for group theory was the quest for solutions of <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a> of degree higher than 4. The 19th-century French mathematician <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a>, extending prior work of <a href="/wiki/Paolo_Ruffini_(mathematician)" class="mw-redirect" title="Paolo Ruffini (mathematician)">Paolo Ruffini</a> and <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>, gave a criterion for the <a href="/wiki/Equation_solving" title="Equation solving">solvability</a> of a particular polynomial equation in terms of the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of its <a href="/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> (solutions). The elements of such a <a href="/wiki/Galois_group" title="Galois group">Galois group</a> correspond to certain <a href="/wiki/Permutation" title="Permutation">permutations</a> of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously.<a href="#cite_note-FOOTNOTEGalois1908-13">[12]</a><a href="#cite_note-FOOTNOTEKleiner1986202-14">[13]</a> More general permutation groups were investigated in particular by <a href="/wiki/Augustin_Louis_Cauchy" class="mw-redirect" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a>. <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a>'s On the theory of groups, as depending on the symbolic equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ^{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta ^{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f208d158b8cf974883449752645a92f283559ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.57ex; height:2.343ex;" alt="{\displaystyle \theta ^{n}=1}" /> (1854) gives the first abstract definition of a <a href="/wiki/Finite_group" title="Finite group">finite group</a>.<a href="#cite_note-FOOTNOTECayley1889-15">[14]</a> Geometry was a second field in which groups were used systematically, especially symmetry groups as part of <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>'s 1872 <a href="/wiki/Erlangen_program" title="Erlangen program">Erlangen program</a>.<a href="#cite_note-FOOTNOTEWussing2007§III.2-16">[15]</a> After novel geometries such as <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic</a> and <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a> founded the study of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> in 1884.<a href="#cite_note-FOOTNOTELie1973-17">[16]</a> The third field contributing to group theory was <a href="/wiki/Number_theory" title="Number theory">number theory</a>. Certain abelian group structures had been used implicitly in <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a>'s number-theoretical work <a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a> (1798), and more explicitly by <a href="/wiki/Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a>.<a href="#cite_note-FOOTNOTEKleiner1986204-18">[17]</a> In 1847, <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a> made early attempts to prove <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> by developing <a href="/wiki/Class_group" class="mw-redirect" title="Class group">groups describing factorization</a> into <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>.<a href="#cite_note-FOOTNOTEWussing2007§I.3.4-19">[18]</a> The convergence of these various sources into a uniform theory of groups started with <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a>'s Traité des substitutions et des équations algébriques (1870).<a href="#cite_note-FOOTNOTEJordan1870-20">[19]</a> <a href="/wiki/Walther_von_Dyck" title="Walther von Dyck">Walther von Dyck</a> (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time.<a href="#cite_note-FOOTNOTEvon_Dyck1882-21">[20]</a> As of the 20th century, groups gained wide recognition by the pioneering work of <a href="/wiki/Ferdinand_Georg_Frobenius" title="Ferdinand Georg Frobenius">Ferdinand Georg Frobenius</a> and <a href="/wiki/William_Burnside" title="William Burnside">William Burnside</a> (who worked on <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> of finite groups), <a href="/wiki/Richard_Brauer" title="Richard Brauer">Richard Brauer</a>'s <a href="/wiki/Modular_representation_theory" title="Modular representation theory">modular representation theory</a> and <a href="/wiki/Issai_Schur" title="Issai Schur">Issai Schur</a>'s papers.<a href="#cite_note-FOOTNOTECurtis2003-22">[21]</a> The theory of Lie groups, and more generally <a href="/wiki/Locally_compact_group" title="Locally compact group">locally compact groups</a> was studied by <a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>, <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> and many others.<a href="#cite_note-FOOTNOTEMackey1976-23">[22]</a> Its <a href="/wiki/Algebra" title="Algebra">algebraic</a> counterpart, the theory of <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a>, was first shaped by <a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a> (from the late 1930s) and later by the work of <a href="/wiki/Armand_Borel" title="Armand Borel">Armand Borel</a> and <a href="/wiki/Jacques_Tits" title="Jacques Tits">Jacques Tits</a>.<a href="#cite_note-FOOTNOTEBorel2001-24">[23]</a> The <a href="/wiki/University_of_Chicago" title="University of Chicago">University of Chicago</a>'s 1960–61 Group Theory Year brought together group theorists such as <a href="/wiki/Daniel_Gorenstein" title="Daniel Gorenstein">Daniel Gorenstein</a>, <a href="/wiki/John_G._Thompson" title="John G. Thompson">John G. Thompson</a> and <a href="/wiki/Walter_Feit" title="Walter Feit">Walter Feit</a>, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>, with the final step taken by <a href="/wiki/Michael_Aschbacher" title="Michael Aschbacher">Aschbacher</a> and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of <a href="/wiki/Mathematical_proof" title="Mathematical proof">proof</a> and number of researchers. Research concerning this classification proof is ongoing.<a href="#cite_note-FOOTNOTESolomon2018-25">[24]</a> Group theory remains a highly active mathematical branch,<a href="#cite_note-26">[b]</a> impacting many other fields, as the <a href="#Examples_and_applications">examples below</a> illustrate. <div class="mw-heading mw-heading2"><h2 id="Elementary_consequences_of_the_group_axioms">Elementary consequences of the group axioms</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=7" title="Edit section: Elementary consequences of the group axioms">edit</a>]</div> Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory.<a href="#cite_note-FOOTNOTELedermann19534–5§1.2-27">[25]</a> For example, <a href="/wiki/Mathematical_induction" title="Mathematical induction">repeated</a> applications of the associativity axiom show that the unambiguity of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a043a2f43a5d9e71633bb4fc25885919f4834e5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.593ex; height:2.843ex;" alt="{\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}" /> generalizes to more than three factors. Because this implies that <a href="/wiki/Bracket#Parentheses_in_mathematics" title="Bracket">parentheses</a> can be inserted anywhere within such a series of terms, parentheses are usually omitted.<a href="#cite_note-FOOTNOTELedermann19733§I.1-28">[26]</a> <div class="mw-heading mw-heading3"><h3 id="Uniqueness_of_identity_element">Uniqueness of identity element</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=8" title="Edit section: Uniqueness of identity element">edit</a>]</div> The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> of a group are equal, because the group axioms imply ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e=e\cdot f=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=e\cdot f=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835c68dcc1a26a97b3cfb8b27b0dcdb91f73e96e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.6ex; height:2.509ex;" alt="{\displaystyle e=e\cdot f=f}" />⁠. It is thus customary to speak of the identity element of the group.<a href="#cite_note-FOOTNOTELang200517§II.1-29">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Uniqueness_of_inverses">Uniqueness of inverses</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=9" title="Edit section: Uniqueness of inverses">edit</a>]</div> The group axioms also imply that the inverse of each element is unique. Let a group element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> have both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" /> as inverses. Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}b&=b\cdot e&&{\text{(}}e{\text{ is the identity element)}}\\&=b\cdot (a\cdot c)&&{\text{(}}c{\text{ and }}a{\text{ are inverses of each other)}}\\&=(b\cdot a)\cdot c&&{\text{(associativity)}}\\&=e\cdot c&&{\text{(}}b{\text{ is an inverse of }}a{\text{)}}\\&=c&&{\text{(}}e{\text{ is the identity element and }}b=c{\text{)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>e</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(</mtext> </mrow> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is the identity element)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(</mtext> </mrow> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> are inverses of each other)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>c</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(associativity)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo>⋅</mo> <mi>c</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(</mtext> </mrow> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is an inverse of </mtext> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>c</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(</mtext> </mrow> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is the identity element and </mtext> </mrow> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}b&=b\cdot e&&{\text{(}}e{\text{ is the identity element)}}\\&=b\cdot (a\cdot c)&&{\text{(}}c{\text{ and }}a{\text{ are inverses of each other)}}\\&=(b\cdot a)\cdot c&&{\text{(associativity)}}\\&=e\cdot c&&{\text{(}}b{\text{ is an inverse of }}a{\text{)}}\\&=c&&{\text{(}}e{\text{ is the identity element and }}b=c{\text{)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbc8ef05318b831b5737d536376ca63afc0af2e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:53.739ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}b&=b\cdot e&&{\text{(}}e{\text{ is the identity element)}}\\&=b\cdot (a\cdot c)&&{\text{(}}c{\text{ and }}a{\text{ are inverses of each other)}}\\&=(b\cdot a)\cdot c&&{\text{(associativity)}}\\&=e\cdot c&&{\text{(}}b{\text{ is an inverse of }}a{\text{)}}\\&=c&&{\text{(}}e{\text{ is the identity element and }}b=c{\text{)}}\end{aligned}}}" /></dd></dl> Therefore, it is customary to speak of the inverse of an element.<a href="#cite_note-FOOTNOTELang200517§II.1-29">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Division"> Division</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=10" title="Edit section: Division">edit</a>]</div> Given elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> of a group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, there is a unique solution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> to the equation ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot x=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>x</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot x=b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/287eeadb64ac963840d9d27237db4cdb466a8269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.335ex; height:2.176ex;" alt="{\displaystyle a\cdot x=b}" />⁠, namely ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a57c12f0c8f7570c2bb67a2d1eaeedf434ec01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.239ex; height:2.676ex;" alt="{\displaystyle a^{-1}\cdot b}" />⁠.<a href="#cite_note-30">[c]</a><a href="#cite_note-FOOTNOTEArtin201840-31">[28]</a> It follows that for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f4a9c67f1896ed706ac35c5e143d0726945ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:2.176ex;" alt="{\displaystyle G\to G}" /> that maps each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/737f1a51a3246bf8139d9beac12202318eaaef4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.239ex; height:1.676ex;" alt="{\displaystyle a\cdot x}" /> is a <a href="/wiki/Bijection" title="Bijection">bijection</a>; it is called left multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> or left translation by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠. Similarly, given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" />⁠, the unique solution to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot a=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot a=b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92e4b8c81bf7247e324ec6d020afb67f48ee0688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.335ex; height:2.176ex;" alt="{\displaystyle x\cdot a=b}" /> is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\cdot a^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\cdot a^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd47fcab898735145f9e796d06f80ae537bd355d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.239ex; height:2.676ex;" alt="{\displaystyle b\cdot a^{-1}}" />⁠. For each ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f4a9c67f1896ed706ac35c5e143d0726945ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:2.176ex;" alt="{\displaystyle G\to G}" /> that maps each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9279782b9a04dd69a5339bf801e1cc3fe8564a1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.239ex; height:1.676ex;" alt="{\displaystyle x\cdot a}" /> is a bijection called right multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> or right translation by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠. <div class="mw-heading mw-heading3"><h3 id="Equivalent_definition_with_relaxed_axioms">Equivalent definition with relaxed axioms</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=11" title="Edit section: Equivalent definition with relaxed axioms">edit</a>]</div> The group axioms for identity and inverses may be "weakened" to assert only the existence of a <a href="/wiki/Left_identity" class="mw-redirect" title="Left identity">left identity</a> and <a href="/wiki/Left_inverse_element" class="mw-redirect" title="Left inverse element">left inverses</a>. From these one-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.<a href="#cite_note-FOOTNOTELang20027§I.2-32">[29]</a> In particular, assuming associativity and the existence of a left identity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /> (that is, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot f=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot f=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d046daa90aad630609b01cb8951b9257c6d8e693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.418ex; height:2.509ex;" alt="{\displaystyle e\cdot f=f}" />⁠) and a left inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" /> for each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> (that is, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}\cdot f=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}\cdot f=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f63a8a3376c5464202ac966a730e918f6f26ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.793ex; height:3.009ex;" alt="{\displaystyle f^{-1}\cdot f=e}" />⁠), one can show that every left inverse is also a right inverse of the same element as follows.<a href="#cite_note-FOOTNOTELang20027§I.2-32">[29]</a> Indeed, one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f\cdot f^{-1}&=e\cdot (f\cdot f^{-1})&&{\text{(left identity)}}\\&=((f^{-1})^{-1}\cdot f^{-1})\cdot (f\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot ((f^{-1}\cdot f)\cdot f^{-1})&&{\text{(associativity)}}\\&=(f^{-1})^{-1}\cdot (e\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot f^{-1}&&{\text{(left identity)}}\\&=e&&{\text{(left inverse)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left identity)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left inverse)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(associativity)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>e</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left inverse)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left identity)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left inverse)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f\cdot f^{-1}&=e\cdot (f\cdot f^{-1})&&{\text{(left identity)}}\\&=((f^{-1})^{-1}\cdot f^{-1})\cdot (f\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot ((f^{-1}\cdot f)\cdot f^{-1})&&{\text{(associativity)}}\\&=(f^{-1})^{-1}\cdot (e\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot f^{-1}&&{\text{(left identity)}}\\&=e&&{\text{(left inverse)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c633e0d6873f5baf31c2b1d1903f11aea67f02f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.171ex; width:54.509ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}f\cdot f^{-1}&=e\cdot (f\cdot f^{-1})&&{\text{(left identity)}}\\&=((f^{-1})^{-1}\cdot f^{-1})\cdot (f\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot ((f^{-1}\cdot f)\cdot f^{-1})&&{\text{(associativity)}}\\&=(f^{-1})^{-1}\cdot (e\cdot f^{-1})&&{\text{(left inverse)}}\\&=(f^{-1})^{-1}\cdot f^{-1}&&{\text{(left identity)}}\\&=e&&{\text{(left inverse)}}\end{aligned}}}" /></dd></dl> Similarly, the left identity is also a right identity:<a href="#cite_note-FOOTNOTELang20027§I.2-32">[29]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}f\cdot e&=f\cdot (f^{-1}\cdot f)&&{\text{(left inverse)}}\\&=(f\cdot f^{-1})\cdot f&&{\text{(associativity)}}\\&=e\cdot f&&{\text{(right inverse)}}\\&=f&&{\text{(left identity)}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>f</mi> <mo>⋅</mo> <mi>e</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left inverse)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>f</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(associativity)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(right inverse)</mtext> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mi>f</mi> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>(left identity)</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f\cdot e&=f\cdot (f^{-1}\cdot f)&&{\text{(left inverse)}}\\&=(f\cdot f^{-1})\cdot f&&{\text{(associativity)}}\\&=e\cdot f&&{\text{(right inverse)}}\\&=f&&{\text{(left identity)}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f20583ff8d62d412bee388d005c109a2323ad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:38.282ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}f\cdot e&=f\cdot (f^{-1}\cdot f)&&{\text{(left inverse)}}\\&=(f\cdot f^{-1})\cdot f&&{\text{(associativity)}}\\&=e\cdot f&&{\text{(right inverse)}}\\&=f&&{\text{(left identity)}}\end{aligned}}}" /></dd></dl> These proofs require all three axioms (associativity, existence of left identity and existence of left inverse). For a structure with a looser definition (like a <a href="/wiki/Semigroup" title="Semigroup">semigroup</a>) one may have, for example, that a left identity is not necessarily a right identity. The same result can be obtained by only assuming the existence of a right identity and a right inverse. However, only assuming the existence of a left identity and a right inverse (or vice versa) is not sufficient to define a group. For example, consider the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\{e,f\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>e</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\{e,f\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9c889934860bcef2155f39b57f65cefcdd89094" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.646ex; height:2.843ex;" alt="{\displaystyle G=\{e,f\}}" /> with the operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" /> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot e=f\cdot e=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>e</mi> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <mi>e</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot e=f\cdot e=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10fce20826feae691c93e9da846b32dea51ab7f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.168ex; height:2.509ex;" alt="{\displaystyle e\cdot e=f\cdot e=e}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e\cdot f=f\cdot f=f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <mi>f</mi> <mo>=</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\cdot f=f\cdot f=f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b03ec9d04a9b5c5ec0144e7ba898e251e7ee7a4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.753ex; height:2.509ex;" alt="{\displaystyle e\cdot f=f\cdot f=f}" />⁠. This structure does have a left identity (namely, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" />⁠), and each element has a right inverse (which is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" /> for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccc71a6904c5ab99ecaab1c8ed69e20815d66da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.317ex; height:2.843ex;" alt="{\displaystyle (G,\cdot )}" /> is not a group, since it lacks a right identity. <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=12" title="Edit section: Basic concepts">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">The following sections use <a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">mathematical symbols</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{x,y,z\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{x,y,z\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0505639e0b978d7464c3d2517af97eff378a048d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.045ex; height:2.843ex;" alt="{\displaystyle X=\{x,y,z\}}" /> to denote a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /> containing <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" />⁠, and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" />⁠, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e580967f68f36743e894aa7944f032dda6ea01d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" /> to state that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> is an element of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" />⁠. The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd1e080abef4bbdab67b43819c6431e7561361c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" /> means <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> associating to every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" /> an element of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" />⁠.</div> When studying sets, one uses concepts such as <a href="/wiki/Subset" title="Subset">subset</a>, function, and <a href="/wiki/Quotient_by_an_equivalence_relation" title="Quotient by an equivalence relation">quotient by an equivalence relation</a>. When studying groups, one uses instead <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>, <a href="/wiki/Group_homomorphism" title="Group homomorphism">homomorphisms</a>, and <a href="/wiki/Quotient_group" title="Quotient group">quotient groups</a>. These are the analogues that take the group structure into account.<a href="#cite_note-33">[d]</a> <div class="mw-heading mw-heading3"><h3 id="Group_homomorphisms">Group homomorphisms</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=13" title="Edit section: Group homomorphisms">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphism</a></div> Group homomorphisms<a href="#cite_note-34">[e]</a> are functions that respect group structure; they may be used to relate two groups. A homomorphism from a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ccc71a6904c5ab99ecaab1c8ed69e20815d66da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.317ex; height:2.843ex;" alt="{\displaystyle (G,\cdot )}" /> to a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (H,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (H,*)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca9efa8f60b1acd949386d15ff163c637042933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.069ex; height:2.843ex;" alt="{\displaystyle (H,*)}" /> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :G\to H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :G\to H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ddc373e6983e5340f633ec5d5905c7613d885e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.962ex; height:2.676ex;" alt="{\displaystyle \varphi :G\to H}" /> such that <style data-mw-deduplicate="TemplateStyles:r996643573">.mw-parser-output .block-indent{padding-left:3em;padding-right:0;overflow:hidden}</style><div class="block-indent" style="padding-left: 1.6em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d549be93f2b9f6b012ada50e47c73dabf9cc90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.415ex; height:2.843ex;" alt="{\displaystyle \varphi (a\cdot b)=\varphi (a)*\varphi (b)}" /> for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠.</div> It would be natural to require also that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" /> respect identities, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (1_{G})=1_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (1_{G})=1_{H}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3445061e046109cabdf053da8bf03da168e15a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.968ex; height:2.843ex;" alt="{\displaystyle \varphi (1_{G})=1_{H}}" />⁠, and inverses, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aec17b5682c7f40d7ef695a885471316b0433358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.883ex; height:3.176ex;" alt="{\displaystyle \varphi (a^{-1})=\varphi (a)^{-1}}" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.<a href="#cite_note-FOOTNOTELang200534§II.3-35">[30]</a> The identity homomorphism of a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> is the homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota _{G}:G\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ι</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \iota _{G}:G\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3830680f1307b832c495a8ca55b2596dae37ca0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.552ex; height:2.509ex;" alt="{\displaystyle \iota _{G}:G\to G}" /> that maps each element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> to itself. An inverse homomorphism of a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :G\to H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :G\to H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ddc373e6983e5340f633ec5d5905c7613d885e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.962ex; height:2.676ex;" alt="{\displaystyle \varphi :G\to H}" /> is a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi :H\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi :H\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51c240629fcc8fb1c9e20821c40ca3f13b51cc0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.955ex; height:2.509ex;" alt="{\displaystyle \psi :H\to G}" /> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \circ \varphi =\iota _{G}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mo>∘</mo> <mi>φ</mi> <mo>=</mo> <msub> <mi>ι</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \circ \varphi =\iota _{G}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59e66755df48fa0d2cbe27d894f0512679f5c0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.674ex; height:2.676ex;" alt="{\displaystyle \psi \circ \varphi =\iota _{G}}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi \circ \psi =\iota _{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>∘</mo> <mi>ψ</mi> <mo>=</mo> <msub> <mi>ι</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi \circ \psi =\iota _{H}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4b70108f72b5c3ba56ec84ed4404f947b43613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.841ex; height:2.676ex;" alt="{\displaystyle \varphi \circ \psi =\iota _{H}}" />⁠, that is, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5b96130cea1b2fef19df775bf480bc507e55ccc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.303ex; height:3.176ex;" alt="{\displaystyle \psi {\bigl (}\varphi (g){\bigr )}=g}" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> and such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797f24f4507002af0b5336091f78995e1ef04415" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.749ex; height:3.176ex;" alt="{\displaystyle \varphi {\bigl (}\psi (h){\bigr )}=h}" /> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠. An <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphism</a> is a homomorphism that has an inverse homomorphism; equivalently, it is a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> homomorphism. Groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> are called isomorphic if there exists an isomorphism ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi :G\to H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi :G\to H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ddc373e6983e5340f633ec5d5905c7613d885e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.962ex; height:2.676ex;" alt="{\displaystyle \varphi :G\to H}" />⁠. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> can be obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> simply by renaming its elements according to the function ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" />⁠; then any statement true for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> is true for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠, provided that any specific elements mentioned in the statement are also renamed. The collection of all groups, together with the homomorphisms between them, form a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>, the <a href="/wiki/Category_of_groups" title="Category of groups">category of groups</a>.<a href="#cite_note-FOOTNOTEMac_Lane1998-36">[31]</a> An <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a> homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :G'\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <msup> <mi>G</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :G'\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f955b4a22d0657834ee0f720a69fa37c637ae9ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.275ex; height:2.843ex;" alt="{\displaystyle \phi :G'\to G}" /> factors canonically as an isomorphism followed by an inclusion, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mo>′</mo> </msup> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mi>H</mi> <mo stretchy="false">↪</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a994a8474a85bff642736494fef17581b392cb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.923ex; height:2.843ex;" alt="{\displaystyle G'\;{\stackrel {\sim }{\to }}\;H\hookrightarrow G}" /> for some subgroup ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠ of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠. Injective homomorphisms are the <a href="/wiki/Monomorphism" title="Monomorphism">monomorphisms</a> in the category of groups. <div class="mw-heading mw-heading3"><h3 id="Subgroups">Subgroups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=14" title="Edit section: Subgroups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></div> Informally, a subgroup is a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> contained within a bigger one, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠: it has a subset of the elements of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, with the same operation.<a href="#cite_note-FOOTNOTELang200519§II.1-37">[32]</a> Concretely, this means that the identity element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> must be contained in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠, and whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14e8880a2e4243a2fe5157e574a0547ef3d5d373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{1}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d2d1733d09f02f33694f72b6da94681021e0f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.509ex;" alt="{\displaystyle h_{2}}" /> are both in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠, then so are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}\cdot h_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}\cdot h_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b9070e666e8bd471b01c2011d2721dc21e26a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.466ex; height:2.509ex;" alt="{\displaystyle h_{1}\cdot h_{2}}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{1}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{1}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ee68e4edfac62efa02d8f361f2d805900629ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.672ex; height:3.343ex;" alt="{\displaystyle h_{1}^{-1}}" />⁠, so the elements of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠, equipped with the group operation on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> restricted to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠, indeed form a group. In this case, the inclusion map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\to G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a576996f91fc78c5b765b8753ee558606b2b758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.504ex; height:2.176ex;" alt="{\displaystyle H\to G}" /> is a homomorphism. In the example of symmetries of a square, the identity and the rotations constitute a subgroup ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bbdd42eb5ac20d48ccc607e220a5c942ab9b95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.537ex; height:2.843ex;" alt="{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}" />⁠, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The <a href="/wiki/Subgroup_test" class="mw-redirect" title="Subgroup test">subgroup test</a> provides a <a href="/wiki/Necessary_and_sufficient_conditions" class="mw-redirect" title="Necessary and sufficient conditions">necessary and sufficient condition</a> for a nonempty subset ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠ of a group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠ to be a subgroup: it is sufficient to check that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}\cdot h\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}\cdot h\in H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6130e0c54abde302a55a5d793bf8bc0178865ea1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.373ex; height:3.009ex;" alt="{\displaystyle g^{-1}\cdot h\in H}" /> for all elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠. Knowing a group's <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">subgroups</a> is important in understanding the group as a whole.<a href="#cite_note-38">[f]</a> Given any subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> of a group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, the subgroup <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> consists of all products of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" /> and their inverses. It is the smallest subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> containing ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" />⁠.<a href="#cite_note-FOOTNOTELedermann197339§II.12-39">[33]</a> In the example of symmetries of a square, the subgroup generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> consists of these two elements, the identity element ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {id} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {id} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b4efd567a73033ece13703ad8f0aa53d59ace7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.939ex; height:2.176ex;" alt="{\displaystyle \mathrm {id} }" />⁠, and the element ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fabe1ea51a4c035b678fc6f0264809afa18fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.405ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }=f_{\mathrm {v} }\cdot r_{2}}" />⁠. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup. <div class="mw-heading mw-heading3"><h3 id="Cosets">Cosets</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=15" title="Edit section: Cosets">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Coset" title="Coset">Coset</a></div> In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> determines left and right cosets, which can be thought of as translations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> by an arbitrary group element ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" />⁠. In symbolic terms, the left and right cosets of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠, containing an element ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" />⁠, are <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r996643573" /><div class="block-indent" style="padding-left: 1.6em;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gH=\{g\cdot h\mid h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>H</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>⋅</mo> <mi>h</mi> <mo>∣</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gH=\{g\cdot h\mid h\in H\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d62e63dd85e73197219d71173769cd0d048d468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.917ex; height:2.843ex;" alt="{\displaystyle gH=\{g\cdot h\mid h\in H\}}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Hg=\{h\cdot g\mid h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>g</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo>⋅</mo> <mi>g</mi> <mo>∣</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Hg=\{h\cdot g\mid h\in H\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3012c30e0b5b7d534ddc5f8a0a0fe3a1f5920daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.917ex; height:2.843ex;" alt="{\displaystyle Hg=\{h\cdot g\mid h\in H\}}" />⁠, respectively.<a href="#cite_note-FOOTNOTELang200541§II.4-40">[34]</a></div> The left cosets of any subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> form a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠; that is, the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of all left cosets is equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> and two left cosets are either equal or have an <a href="/wiki/Empty_set" title="Empty set">empty</a> <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>.<a href="#cite_note-FOOTNOTELang200212§I.2-41">[35]</a> The first case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}H=g_{2}H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>H</mi> <mo>=</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}H=g_{2}H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdc4c4347ad22a16c2d269173f33bf9efd85f8d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.552ex; height:2.509ex;" alt="{\displaystyle g_{1}H=g_{2}H}" /> happens <a href="/wiki/If_and_only_if" title="If and only if">precisely when</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{1}^{-1}\cdot g_{2}\in H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>⋅</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{1}^{-1}\cdot g_{2}\in H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657227f032b674f1d7f4b0265895a0b06721108a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.198ex; height:3.343ex;" alt="{\displaystyle g_{1}^{-1}\cdot g_{2}\in H}" />⁠, i.e., when the two elements differ by an element of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠. Similar considerations apply to the right cosets of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" />⁠. The left cosets of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> may or may not be the same as its right cosets. If they are (that is, if all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> satisfy ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gH=Hg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>H</mi> <mo>=</mo> <mi>H</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gH=Hg}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1dec58109c201c75ce8384c01a1aba2c6fcd03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.458ex; height:2.509ex;" alt="{\displaystyle gH=Hg}" />⁠), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> is said to be a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>. In ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠, the group of symmetries of a square, with its subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> of rotations, the left cosets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gR}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad2c861c51f57c1a74d5a70fc181a944691b588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle gR}" /> are either equal to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" />⁠, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /> is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> itself, or otherwise equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c01208b07a7f1cee56acda5b09ef193d7c46df1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.184ex; height:2.843ex;" alt="{\displaystyle U=f_{\mathrm {c} }R=\{f_{\mathrm {c} },f_{\mathrm {d} },f_{\mathrm {v} },f_{\mathrm {h} }\}}" /> (highlighted in green in the Cayley table of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠). The subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> is normal, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>=</mo> <mi>U</mi> <mo>=</mo> <mi>R</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f1ee356a3bfeef5755f0983975fb0a5d977189" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.711ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {c} }R=U=Rf_{\mathrm {c} }}" /> and similarly for the other elements of the group. (In fact, in the case of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" />⁠, the cosets generated by reflections are all equal: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">h</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d87d20063ecff019abdee9067b05d7280367720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.263ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {h} }R=f_{\mathrm {v} }R=f_{\mathrm {d} }R=f_{\mathrm {c} }R}" />⁠.) <div class="mw-heading mw-heading3"><h3 id="Quotient_groups">Quotient groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=16" title="Edit section: Quotient groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></div> Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> is a normal subgroup of a group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N=\{gN\mid g\in G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mi>N</mi> <mo>∣</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N=\{gN\mid g\in G\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a44144e8ea65a87623f26ecdca0be134e932e3c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.376ex; height:2.843ex;" alt="{\displaystyle G/N=\{gN\mid g\in G\}}" /> denotes its set of cosets. Then there is a unique group law on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}" /> for which the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\to G/N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c203e099a7f440cb370e1f0714c364473178a17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.494ex; height:2.843ex;" alt="{\displaystyle G\to G/N}" /> sending each element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd554e0bd9bf40da2dac81a62a5c954d56d3aa74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.18ex; height:2.509ex;" alt="{\displaystyle gN}" /> is a homomorphism. Explicitly, the product of two cosets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd554e0bd9bf40da2dac81a62a5c954d56d3aa74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.18ex; height:2.509ex;" alt="{\displaystyle gN}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle hN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle hN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1567ea30a2b51f7f51c26013837f478cdc8b92ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.403ex; height:2.176ex;" alt="{\displaystyle hN}" /> is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (gh)N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (gh)N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e0696475ecede9efe9bfc891b182b36d78b5946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.328ex; height:2.843ex;" alt="{\displaystyle (gh)N}" />⁠, the coset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle eN=N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mi>N</mi> <mo>=</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle eN=N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392c569f369e3f2b35d06ff6822f18189a7da05d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.309ex; height:2.176ex;" alt="{\displaystyle eN=N}" /> serves as the identity of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}" />⁠, and the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd554e0bd9bf40da2dac81a62a5c954d56d3aa74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.18ex; height:2.509ex;" alt="{\displaystyle gN}" /> in the quotient group is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mi>N</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb5b8b1629f0c952bdab036c12fe5d79f0b27443" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.452ex; height:3.343ex;" alt="{\displaystyle (gN)^{-1}=\left(g^{-1}\right)N}" />⁠. The group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}" />⁠, read as "⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠ modulo ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" />⁠",<a href="#cite_note-FOOTNOTELang200545§II.4-42">[36]</a> is called a quotient group or factor group. The quotient group can alternatively be characterized by a <a href="/wiki/Universal_property" title="Universal property">universal property</a>. <table class="wikitable" style="float:right; text-align:center; margin:.5em 0 .5em 1em; width:200px;"> <caption>Cayley table of the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}/R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}/R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1432bb683f71bda9c8734946a3544034bf8084eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.756ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} _{4}/R}" /> </caption> <tbody><tr> <th style="width:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" /> </th> <th style="width:33%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> </th> <th style="width:33%;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> </th></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> </td></tr> <tr> <th><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" /></td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> </td></tr></tbody></table> The elements of the quotient group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}/R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}/R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1432bb683f71bda9c8734946a3544034bf8084eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.756ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} _{4}/R}" /> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=f_{\mathrm {v} }R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=f_{\mathrm {v} }R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a407438ca730be06dbd72a825224cc3e2831132" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.885ex; height:2.509ex;" alt="{\displaystyle U=f_{\mathrm {v} }R}" />⁠. The group operation on the quotient is shown in the table. For example, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>⋅</mo> <mi>U</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mi>R</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mi>R</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aefbf77a4dce7cffe06283ec209c51bdbef2e6a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.721ex; height:2.843ex;" alt="{\displaystyle U\cdot U=f_{\mathrm {v} }R\cdot f_{\mathrm {v} }R=(f_{\mathrm {v} }\cdot f_{\mathrm {v} })R=R}" />⁠. Both the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">d</mi> </mrow> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bbdd42eb5ac20d48ccc607e220a5c942ab9b95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.537ex; height:2.843ex;" alt="{\displaystyle R=\{\mathrm {id} ,r_{1},r_{2},r_{3}\}}" /> and the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}/R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}/R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1432bb683f71bda9c8734946a3544034bf8084eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.756ex; height:2.843ex;" alt="{\displaystyle \mathrm {D} _{4}/R}" /> are abelian, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> construction; <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> is an example. The <a href="/wiki/First_isomorphism_theorem" class="mw-redirect" title="First isomorphism theorem">first isomorphism theorem</a> implies that any <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :G\to H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :G\to H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39c7a655e5848542428a585814efb99648f189d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.827ex; height:2.509ex;" alt="{\displaystyle \phi :G\to H}" /> factors canonically as a quotient homomorphism followed by an isomorphism: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ker</mi> <mo>⁡</mo> <mi>ϕ</mi> <mspace width="thickmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∼</mo> </mrow> </mover> </mrow> </mrow> <mspace width="thickmathspace"></mspace> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d46547528125b77a6513d711d0d42e727744dac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.439ex; height:3.343ex;" alt="{\displaystyle G\to G/\ker \phi \;{\stackrel {\sim }{\to }}\;H}" />⁠. Surjective homomorphisms are the <a href="/wiki/Epimorphism" title="Epimorphism">epimorphisms</a> in the category of groups. <div class="mw-heading mw-heading3"><h3 id="Presentations">Presentations</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=17" title="Edit section: Presentations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">Presentation of a group</a></div> Every group is isomorphic to a quotient of a <a href="/wiki/Free_group" title="Free group">free group</a>, in many ways. For example, the dihedral group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> is generated by the right rotation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> and the reflection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> in a vertical line (every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> is a finite product of copies of these and their inverses). Hence there is a surjective homomorphism ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" />⁠ from the free group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r,f\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>r</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle r,f\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cc10d40eaf2d77ac0c4b4ca5c0b9d1655a38b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.17ex; height:2.843ex;" alt="{\displaystyle \langle r,f\rangle }" /> on two generators to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> sending <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" /> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /> to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50dfd257a51e037112c917f8a9e47c9c053466df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.509ex;" alt="{\displaystyle f_{1}}" />⁠. Elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/770f083d98de247a8c5e1e0df32fb0c3a9806286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.944ex; height:2.509ex;" alt="{\displaystyle \ker \phi }" /> are called relations; examples include ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{4},f^{2},(r\cdot f)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{4},f^{2},(r\cdot f)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d567d05938057138cea3b5f20ce2325661a302c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.415ex; height:3.176ex;" alt="{\displaystyle r^{4},f^{2},(r\cdot f)^{2}}" />⁠. In fact, it turns out that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/770f083d98de247a8c5e1e0df32fb0c3a9806286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.944ex; height:2.509ex;" alt="{\displaystyle \ker \phi }" /> is the smallest normal subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r,f\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>r</mi> <mo>,</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle r,f\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7cc10d40eaf2d77ac0c4b4ca5c0b9d1655a38b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.17ex; height:2.843ex;" alt="{\displaystyle \langle r,f\rangle }" /> containing these three elements; in other words, all relations are consequences of these three. The quotient of the free group by this normal subgroup is denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>r</mi> <mo>,</mo> <mi>f</mi> <mo>∣</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b70df00a194b8b9395c6a9d21a3cf0c3811c03f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.913ex; height:3.176ex;" alt="{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle }" />⁠. This is called a <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> by generators and relations, because the first isomorphism theorem for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" />⁠ yields an isomorphism ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>r</mi> <mo>,</mo> <mi>f</mi> <mo>∣</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>⋅</mo> <mi>f</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">→</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c1ed95febec35364b47dfac294eff2b1799c57e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.357ex; height:3.176ex;" alt="{\displaystyle \langle r,f\mid r^{4}=f^{2}=(r\cdot f)^{2}=1\rangle \to \mathrm {D} _{4}}" />⁠.<a href="#cite_note-FOOTNOTELang20029§I.2-43">[37]</a> A presentation of a group can be used to construct the <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a>, a graphical depiction of a <a href="/wiki/Discrete_group" title="Discrete group">discrete group</a>.<a href="#cite_note-FOOTNOTEMagnusKarrassSolitar200456–67§1.6-44">[38]</a> <div class="mw-heading mw-heading2"><h2 id="Examples_and_applications">Examples and applications</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=18" title="Edit section: Examples and applications">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wallpaper_group-cm-6.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group-cm-6.jpg/220px-Wallpaper_group-cm-6.jpg" decoding="async" width="220" height="249" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group-cm-6.jpg/330px-Wallpaper_group-cm-6.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Wallpaper_group-cm-6.jpg/440px-Wallpaper_group-cm-6.jpg 2x" data-file-width="1132" data-file-height="1282" /></a><figcaption>A periodic wallpaper pattern gives rise to a <a href="/wiki/Wallpaper_group" title="Wallpaper group">wallpaper group</a>.</figcaption></figure> Examples and applications of groups abound. A starting point is the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /> of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative groups</a>. These groups are predecessors of important constructions in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>. Groups are also applied in many other mathematical areas. Mathematical objects are often examined by <a href="/wiki/Functor" title="Functor">associating</a> groups to them and studying the properties of the corresponding groups. For example, <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a> founded what is now called <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a> by introducing the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a>.<a href="#cite_note-FOOTNOTEHatcher200230Chapter_I-45">[39]</a> By means of this connection, <a href="/wiki/Glossary_of_topology" class="mw-redirect" title="Glossary of topology">topological properties</a> such as <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">proximity</a> and <a href="/wiki/Continuous_function" title="Continuous function">continuity</a> translate into properties of groups.<a href="#cite_note-46">[g]</a> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fundamental_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Fundamental_group.svg/220px-Fundamental_group.svg.png" decoding="async" width="220" height="209" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Fundamental_group.svg/330px-Fundamental_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Fundamental_group.svg/440px-Fundamental_group.svg.png 2x" data-file-width="305" data-file-height="290" /></a><figcaption>The fundamental group of a plane minus a point (bold) consists of loops around the missing point. This group is isomorphic to the integers under addition.</figcaption></figure> Elements of the fundamental group of a <a href="/wiki/Topological_space" title="Topological space">topological space</a> are <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of loops, where loops are considered equivalent if one can be <a href="/wiki/Homotopy" title="Homotopy">smoothly deformed</a> into another, and the group operation is "concatenation" (tracing one loop then the other). For example, as shown in the figure, if the topological space is the plane with one point removed, then loops which do not wrap around the missing point (blue) <a href="/wiki/Null-homotopic" class="mw-redirect" title="Null-homotopic">can be smoothly contracted to a single point</a> and are the identity element of the fundamental group. A loop which wraps around the missing point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /> times cannot be deformed into a loop which wraps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /> times (with ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\neq k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>≠</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\neq k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0bf29e7425f390fb83e7839d7a794f9d903b57b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.35ex; height:2.676ex;" alt="{\displaystyle m\neq k}" />⁠), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by its <a href="/wiki/Winding_number" title="Winding number">winding number</a> around the missing point. The resulting group is isomorphic to the integers under addition. In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.<a href="#cite_note-47">[h]</a> In a similar vein, <a href="/wiki/Geometric_group_theory" title="Geometric group theory">geometric group theory</a> employs geometric concepts, for example in the study of <a href="/wiki/Hyperbolic_group" title="Hyperbolic group">hyperbolic groups</a>.<a href="#cite_note-FOOTNOTECoornaertDelzantPapadopoulos1990-48">[40]</a> Further branches crucially applying groups include <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and number theory.<a href="#cite_note-49">[41]</a> In addition to the above theoretical applications, many practical applications of groups exist. <a href="/wiki/Cryptography" title="Cryptography">Cryptography</a> relies on the combination of the abstract group theory approach together with <a href="/wiki/Algorithm" title="Algorithm">algorithmical</a> knowledge obtained in <a href="/wiki/Computational_group_theory" title="Computational group theory">computational group theory</a>, in particular when implemented for finite groups.<a href="#cite_note-FOOTNOTESeress1997-50">[42]</a> Applications of group theory are not restricted to mathematics; sciences such as <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> and <a href="/wiki/Computer_science" title="Computer science">computer science</a> benefit from the concept. <div class="mw-heading mw-heading3"><h3 id="Numbers">Numbers</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=19" title="Edit section: Numbers">edit</a>]</div> Many number systems, such as the integers and the <a href="/wiki/Rational_number" title="Rational number">rationals</a>, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> and fields. Further abstract algebraic concepts such as <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>, <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> and <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebras</a> also form groups. <div class="mw-heading mw-heading4"><h4 id="Integers">Integers</h4>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=20" title="Edit section: Integers">edit</a>]</div> The group of integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /> under addition, denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Z} ,+\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Z} ,+\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/879fc588c23501c556ac2e23eee96178857acdaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.202ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Z} ,+\right)}" />⁠, has been described above. The integers, with the operation of multiplication instead of addition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Z} ,\cdot \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>⋅</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Z} ,\cdot \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faceb8696dfd3fa47e625c719fef73d9f0add6c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.04ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Z} ,\cdot \right)}" /> do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4208bf5a67fc2ceb3a3bcd75aebb1d74fbb531bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=2}" /> is an integer, but the only solution to the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/943794bd7ed5a52298d3c7e44eceea37724e2d1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.167ex; height:2.176ex;" alt="{\displaystyle a\cdot b=1}" /> in this case is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b={\tfrac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b={\tfrac {1}{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10c679e86a30f96109e39c3f39fb1aff84a7c171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.754ex; height:3.509ex;" alt="{\displaystyle b={\tfrac {1}{2}}}" />⁠, which is a rational number, but not an integer. Hence not every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" /> has a (multiplicative) inverse.<a href="#cite_note-51">[i]</a> <div class="mw-heading mw-heading4"><h4 id="Rationals">Rationals</h4>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=21" title="Edit section: Rationals">edit</a>]</div> The desire for the existence of multiplicative inverses suggests considering <a href="/wiki/Fraction_(mathematics)" class="mw-redirect" title="Fraction (mathematics)">fractions</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{b}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{b}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ca6e3e42a0093600ae2abfd5587ae52136547a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.713ex; height:4.843ex;" alt="{\displaystyle {\frac {a}{b}}.}" /> Fractions of integers (with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> nonzero) are known as <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>.<a href="#cite_note-52">[j]</a> The set of all such irreducible fractions is commonly denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" />⁠. There is still a minor obstacle for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Q} ,\cdot \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mo>⋅</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Q} ,\cdot \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2397d850c8a0516cecffad661694f37261671fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Q} ,\cdot \right)}" />⁠, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /> such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot 0=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot 0=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced816271f5dd84e7470eb0452c4d61482648ad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.432ex; height:2.176ex;" alt="{\displaystyle x\cdot 0=1}" />⁠), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Q} ,\cdot \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>,</mo> <mo>⋅</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Q} ,\cdot \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2397d850c8a0516cecffad661694f37261671fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Q} ,\cdot \right)}" /> is still not a group. However, the set of all nonzero rational numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∖</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>}</mo> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>q</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∣</mo> <mi>q</mi> <mo>≠</mo> <mn>0</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11988f50208f63e2ed07fe71e83c0c563d2eccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.545ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} \smallsetminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}" /> does form an abelian group under multiplication, also denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} ^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e97ecc516980bf3841b50135e28952a1b25a9127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.676ex;" alt="{\displaystyle \mathbb {Q} ^{\times }}" />⁠.<a href="#cite_note-53">[k]</a> Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a/b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a/b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1b6c014398323cb45578581a536033fce1b28c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.39ex; height:2.843ex;" alt="{\displaystyle a/b}" /> is ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b/a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e0082708c4a962619f1b493a22dc776b5fbab62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.39ex; height:2.843ex;" alt="{\displaystyle b/a}" />⁠, therefore the axiom of the inverse element is satisfied. The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> by other than zero is possible, such as in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" /> – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.<a href="#cite_note-54">[l]</a> <div class="mw-heading mw-heading3"><h3 id="Modular_arithmetic">Modular arithmetic</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=22" title="Edit section: Modular arithmetic">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Clock_group.svg" class="mw-file-description"><img alt="The clock hand points to 9 o'clock; 4 hours later it is at 1 o'clock." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/250px-Clock_group.svg.png" decoding="async" width="220" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/330px-Clock_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Clock_group.svg/500px-Clock_group.svg.png 2x" data-file-width="560" data-file-height="245" /></a><figcaption>The hours on a clock form a group that uses <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">addition modulo</a> 12. Here, 9 + 4 ≡ 1.</figcaption></figure> Modular arithmetic for a modulus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> defines any two elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> that differ by a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> to be equivalent, denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\equiv b{\pmod {n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv b{\pmod {n}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbb66ad4d03232b185b3dd6a6ee293943f21f786" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.405ex; height:2.843ex;" alt="{\displaystyle a\equiv b{\pmod {n}}}" />⁠. Every integer is equivalent to one of the integers from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" />⁠, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent <a href="/wiki/Representative_(mathematics)" class="mw-redirect" title="Representative (mathematics)">representative</a>. Modular addition, defined in this way for the integers from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd0b0f32b28f51962943ee9ede4fb34198a2521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" />⁠, forms a group, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Z} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1519d0155077363f01706851836d19c03b4b2542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.639ex; height:2.509ex;" alt="{\displaystyle \mathrm {Z} _{n}}" /> or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494b84eb357d1080370c7474fb3fd99e17f2bc2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.309ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}" />⁠, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" /> as the identity element and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/316566d138f897647f6c6e7db9af4a17af4dc063" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.465ex; height:2.176ex;" alt="{\displaystyle n-a}" /> as the inverse element of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠. A familiar example is addition of hours on the face of a <a href="/wiki/12-hour_clock" title="12-hour clock">clock</a>, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d3d1e1f9dfe0254c628379e69a69711fe4eabd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 9}" /> and is advanced <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}" /> hours, it ends up on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠, as shown in the illustration. This is expressed by saying that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9+4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>+</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9+4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a186f7282b89880bd3bf5dd8f36ff917a5e25474" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 9+4}" /> is congruent to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> "modulo ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a522d3aa5812a136a69f06e1b909d809e849be39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 12}" />⁠" or, in symbols, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9+4\equiv 1{\pmod {12}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>+</mo> <mn>4</mn> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mn>12</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9+4\equiv 1{\pmod {12}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da4e61245d43aea3d651905502a019b67b47e66" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.082ex; height:2.843ex;" alt="{\displaystyle 9+4\equiv 1{\pmod {12}}.}" /> For any prime number ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />⁠, there is also the <a href="/wiki/Multiplicative_group_of_integers_modulo_n" title="Multiplicative group of integers modulo n">multiplicative group of integers modulo ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />⁠</a>.<a href="#cite_note-FOOTNOTELang2005Chapter_VII-55">[43]</a> Its elements can be represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f356ae51988add41a7da343e6b6d48fa968da162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p-1}" />⁠. The group operation, multiplication modulo ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />⁠, replaces the usual product by its representative, the <a href="/wiki/Remainder" title="Remainder">remainder</a> of division by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />⁠. For example, for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261ae92684f7423f53230594c6aca12906781b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=5}" />⁠, the four group elements can be represented by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,3,4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,3,4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be93d63fddb04c779e9c815b99a82aa8073f577f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.752ex; height:2.509ex;" alt="{\displaystyle 1,2,3,4}" />⁠. In this group, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mn>4</mn> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48e08d7a2d3eb447792c870d464db03c4b893b49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.109ex; height:2.176ex;" alt="{\displaystyle 4\cdot 4\equiv 1{\bmod {5}}}" />⁠, because the usual product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/960615e346e1c003a911f45b1225113ea01b4ff7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 16}" /> is equivalent to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠: when divided by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}" /> it yields a remainder of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠. The primality of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> ensures that the usual product of two representatives is not divisible by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />⁠, and therefore that the modular product is nonzero.<a href="#cite_note-56">[m]</a> The identity element is represented by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> not divisible by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />⁠, there exists an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b\equiv 1{\pmod {p}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em"></mspace> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em"></mspace> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b\equiv 1{\pmod {p}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d713089c0936afdad8e4a300bd9271a847d2d61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.668ex; height:2.843ex;" alt="{\displaystyle a\cdot b\equiv 1{\pmod {p}},}" /> that is, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> evenly divides ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5fe719429af1511db62a45e28f974348fc1dc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.909ex; height:2.343ex;" alt="{\displaystyle a\cdot b-1}" />⁠. The inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" /> can be found by using <a href="/wiki/B%C3%A9zout%27s_identity" title="Bézout's identity">Bézout's identity</a> and the fact that the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(a,p)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">gcd</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gcd(a,p)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfbdb155824dbbecdaa958488109673bf60f5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.73ex; height:2.843ex;" alt="{\displaystyle \gcd(a,p)}" /> equals ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠.<a href="#cite_note-FOOTNOTERosen200054&nbsp;(Theorem_2.1)-57">[44]</a> In the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261ae92684f7423f53230594c6aca12906781b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=5}" /> above, the inverse of the element represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}" /> is that represented by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}" />⁠, and the inverse of the element represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" /> is represented by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}" />⁠, as ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f54c412404a510c711481295bcdb153deab6fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:19.369ex; height:2.176ex;" alt="{\displaystyle 3\cdot 2=6\equiv 1{\bmod {5}}}" />⁠. Hence all group axioms are fulfilled. This example is similar to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∖</mo> <mrow> <mo>{</mo> <mn>0</mn> <mo>}</mo> </mrow> <mo>,</mo> <mo>⋅</mo> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c37a587031f0d600726b613c1fe34ced87437054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.013ex; height:2.843ex;" alt="{\displaystyle \left(\mathbb {Q} \smallsetminus \left\{0\right\},\cdot \right)}" /> above: it consists of exactly those elements in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57869a3a3c4c431cc49c4c7ab1d9c7ea692b517b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.433ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p\mathbb {Z} }" /> that have a multiplicative inverse.<a href="#cite_note-FOOTNOTELang2005292§VIII.1-58">[45]</a> These groups, denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46b809ee9c571d5b49222a9d28c76062a3c3414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.931ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{p}^{\times }}" />⁠, are crucial to <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>.<a href="#cite_note-59">[n]</a> <div class="mw-heading mw-heading3"><h3 id="Cyclic_groups">Cyclic groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=23" title="Edit section: Cyclic groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img alt="A hexagon whose corners are located regularly on a circle" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/170px-Cyclic_group.svg.png" decoding="async" width="170" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/255px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/340px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a><figcaption>The 6th complex roots of unity form a cyclic group. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /> is a primitive element, but <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f23e4338924d253cd2f3ed83a7082fb07243d08e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="{\displaystyle z^{2}}" /> is not, because the odd powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /> are not a power of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f23e4338924d253cd2f3ed83a7082fb07243d08e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.145ex; height:2.676ex;" alt="{\displaystyle z^{2}}" />⁠.</figcaption></figure> A cyclic group is a group all of whose elements are <a href="/wiki/Power_(mathematics)" class="mw-redirect" title="Power (mathematics)">powers</a> of a particular element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />.<a href="#cite_note-FOOTNOTELang200522§II.1-60">[46]</a> In multiplicative notation, the elements of the group are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> <mi>a</mi> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85884885f71fb1f6086f56b1dd381c9527c02b0b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.909ex; height:3.009ex;" alt="{\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^{0},a,a^{2},a^{3},\dots ,}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f564e5dc0b6e68af32ca8614e972f5b36e944a24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.284ex; height:2.676ex;" alt="{\displaystyle a^{2}}" /> means ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ff23a152fce8b376b88b45e3eea0a724a9ebb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.139ex; height:1.676ex;" alt="{\displaystyle a\cdot a}" />⁠, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1578ece74213cf0fc591f887180a47bb735df67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-3}}" /> stands for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63398fd9d26cc41ce6ef3e3310055d0d90f37d6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.334ex; height:3.176ex;" alt="{\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a)^{-1}}" />⁠, etc.<a href="#cite_note-61">[o]</a> Such an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> is called a generator or a <a href="/wiki/Primitive_root_modulo_n" title="Primitive root modulo n">primitive element</a> of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>a</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mo>…</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b4a339914a56e51862367a8165a75e4a1f2045" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.336ex; height:2.843ex;" alt="{\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}" /> In the groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494b84eb357d1080370c7474fb3fd99e17f2bc2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.309ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /n\mathbb {Z} ,+)}" /> introduced above, the element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" /> is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" />⁠. Any cyclic group with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> elements is isomorphic to this group. A second example for cyclic groups is the group of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />⁠th <a href="/wiki/Root_of_unity" title="Root of unity">complex roots of unity</a>, given by <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /> satisfying ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fb31edf665701c275c44a5be8b82a95509888d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.57ex; height:2.343ex;" alt="{\displaystyle z^{n}=1}" />⁠. These numbers can be visualized as the <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> on a regular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-gon, as shown in blue in the image for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}" />⁠. The group operation is multiplication of complex numbers. In the picture, multiplying with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /> corresponds to a <a href="/wiki/Clockwise" title="Clockwise">counter-clockwise</a> rotation by 60°.<a href="#cite_note-FOOTNOTELang200526§II.2-62">[47]</a> From <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>, the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46b809ee9c571d5b49222a9d28c76062a3c3414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.931ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{p}^{\times }}" /> is cyclic for prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" />: for example, if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261ae92684f7423f53230594c6aca12906781b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=5}" />⁠, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}" /> is a generator since ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{1}=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{1}=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65456524edacf89dc8c4e3f7872f1cbed93e7130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 3^{1}=3}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}=9\equiv 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>9</mn> <mo>≡</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}=9\equiv 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a09cda133382b9545be314eae8ad1a2a92bb6cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.739ex; height:2.676ex;" alt="{\displaystyle 3^{2}=9\equiv 4}" />⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{3}\equiv 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>≡</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{3}\equiv 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abe821a0f1d674d0a9877ba1c1c6d4bc102aad54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 3^{3}\equiv 2}" />⁠, and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{4}\equiv 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>≡</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{4}\equiv 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fac284c4f392b80a6874481b7e51d81437e80a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.478ex; height:2.676ex;" alt="{\displaystyle 3^{4}\equiv 1}" />⁠. Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠, all the powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/910eaae0a8267ccb04d4846f6a28f02ce6ab8ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.202ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} ,+)}" />⁠, the group of integers under addition introduced above.<a href="#cite_note-FOOTNOTELang200522§II.1_(example_11)-63">[48]</a> As these two prototypes are both abelian, so are all cyclic groups. The study of finitely generated abelian groups is quite mature, including the <a href="/wiki/Fundamental_theorem_of_finitely_generated_abelian_groups" class="mw-redirect" title="Fundamental theorem of finitely generated abelian groups">fundamental theorem of finitely generated abelian groups</a>; and reflecting this state of affairs, many group-related notions, such as <a href="/wiki/Center_(group_theory)" title="Center (group theory)">center</a> and <a href="/wiki/Commutator" title="Commutator">commutator</a>, describe the extent to which a given group is not abelian.<a href="#cite_note-FOOTNOTELang200226,_29§I.5-64">[49]</a> <div class="mw-heading mw-heading3"><h3 id="Symmetry_groups">Symmetry groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=24" title="Edit section: Symmetry groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Symmetry_group" title="Symmetry group">Symmetry group</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Molecular_symmetry" title="Molecular symmetry">Molecular symmetry</a>, <a href="/wiki/Space_group" title="Space group">Space group</a>, <a href="/wiki/Point_group" title="Point group">Point group</a>, and <a href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">Symmetry in physics</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Uniform_tiling_73-t2_colored.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Uniform_tiling_73-t2_colored.png/250px-Uniform_tiling_73-t2_colored.png" decoding="async" width="170" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Uniform_tiling_73-t2_colored.png/330px-Uniform_tiling_73-t2_colored.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Uniform_tiling_73-t2_colored.png/500px-Uniform_tiling_73-t2_colored.png 2x" data-file-width="935" data-file-height="930" /></a><figcaption>The (2,3,7) triangle group, a hyperbolic reflection group, acts on this <a href="/wiki/Tessellation" title="Tessellation">tiling</a> of the <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic</a> plane<a href="#cite_note-FOOTNOTEEllis2019-65">[50]</a></figcaption></figure> Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below).<a href="#cite_note-FOOTNOTEWeyl1952-66">[51]</a> Conceptually, group theory can be thought of as the study of symmetry.<a href="#cite_note-67">[p]</a> <a href="/wiki/Symmetry_in_mathematics" title="Symmetry in mathematics">Symmetries in mathematics</a> greatly simplify the study of <a href="/wiki/Geometry" title="Geometry">geometrical</a> or <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analytical</a> objects. A group is said to <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">act</a> on another mathematical object ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" />⁠ if every group element can be associated to some operation on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" />⁠ and the composition of these operations follows the group law. For example, an element of the <a href="/wiki/(2,3,7)_triangle_group" title="(2,3,7) triangle group">(2,3,7) triangle group</a> acts on a triangular <a href="/wiki/Tessellation" title="Tessellation">tiling</a> of the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a> by permuting the triangles.<a href="#cite_note-FOOTNOTEEllis2019-65">[50]</a> By a group action, the group pattern is connected to the structure of the object being acted on. In chemistry, <a href="/wiki/Point_group" title="Point group">point groups</a> describe <a href="/wiki/Molecular_symmetry" title="Molecular symmetry">molecular symmetries</a>, while <a href="/wiki/Space_group" title="Space group">space groups</a> describe crystal symmetries in <a href="/wiki/Crystallography" title="Crystallography">crystallography</a>. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanical</a> analysis of these properties.<a href="#cite_note-68">[52]</a> For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.<a href="#cite_note-FOOTNOTEWeyl1950197–202-69">[53]</a> Group theory helps predict the changes in physical properties that occur when a material undergoes a <a href="/wiki/Phase_transition" title="Phase transition">phase transition</a>, for example, from a cubic to a tetrahedral crystalline form. An example is <a href="/wiki/Ferroelectric" class="mw-redirect" title="Ferroelectric">ferroelectric</a> materials, where the change from a paraelectric to a ferroelectric state occurs at the <a href="/wiki/Curie_temperature" title="Curie temperature">Curie temperature</a> and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft <a href="/wiki/Phonon" title="Phonon">phonon</a> mode, a vibrational lattice mode that goes to zero frequency at the transition.<a href="#cite_note-FOOTNOTEDove2003-70">[54]</a> Such <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">spontaneous symmetry breaking</a> has found further application in elementary particle physics, where its occurrence is related to the appearance of <a href="/wiki/Goldstone_boson" title="Goldstone boson">Goldstone bosons</a>.<a href="#cite_note-FOOTNOTEZee2010228-71">[55]</a> <table class="wikitable" style="text-align:center; margin:1em auto 1em auto;"> <tbody><tr> <td width="20%"><a href="/wiki/File:C60_Molecule.svg" class="mw-file-description"><img alt="A schematic depiction of a Buckminsterfullerene molecule" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/C60_Molecule.svg/250px-C60_Molecule.svg.png" decoding="async" width="125" height="123" class="mw-file-element" data-file-width="576" data-file-height="566" /></a> </td> <td width="25%"><a href="/wiki/File:Ammonia-3D-balls-A.png" class="mw-file-description"><img alt="A schematic depiction of an Ammonia molecule" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Ammonia-3D-balls-A.png/250px-Ammonia-3D-balls-A.png" decoding="async" width="125" height="96" class="mw-file-element" data-file-width="1100" data-file-height="849" /></a> </td> <td width="15%"><a href="/wiki/File:Cubane-3D-balls.png" class="mw-file-description"><img alt="A schematic depiction of a cubane molecule" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Cubane-3D-balls.png/125px-Cubane-3D-balls.png" decoding="async" width="125" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Cubane-3D-balls.png/188px-Cubane-3D-balls.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Cubane-3D-balls.png/250px-Cubane-3D-balls.png 2x" data-file-width="1100" data-file-height="1111" /></a> </td> <td width="20%"><a href="/wiki/File:K2PtCl4.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/K2PtCl4.png/250px-K2PtCl4.png" decoding="async" width="125" height="46" class="mw-file-element" data-file-width="542" data-file-height="198" /></a> </td></tr> <tr> <td><a href="/wiki/Buckminsterfullerene" title="Buckminsterfullerene">Buckminsterfullerene</a> displays <a href="/wiki/Icosahedral_symmetry" title="Icosahedral symmetry">icosahedral symmetry</a><a href="#cite_note-FOOTNOTEChanceyO'Brien202115,_16-72">[56]</a> </td> <td><a href="/wiki/Ammonia" title="Ammonia">Ammonia</a>, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.<a href="#cite_note-FOOTNOTESimons2003§4.2.1-73">[57]</a> </td> <td><a href="/wiki/Cubane" title="Cubane">Cubane</a> C8H8 features <a href="/wiki/Octahedral_symmetry" title="Octahedral symmetry">octahedral symmetry</a>.<a href="#cite_note-FOOTNOTEElielWilenMander199482-74">[58]</a> </td> <td>The <a href="/wiki/Potassium_tetrachloroplatinate" title="Potassium tetrachloroplatinate">tetrachloroplatinate(II)</a> ion, [PtCl4]2− exhibits square-planar geometry </td></tr></tbody></table> Finite symmetry groups such as the <a href="/wiki/Mathieu_group" title="Mathieu group">Mathieu groups</a> are used in <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>, which is in turn applied in <a href="/wiki/Forward_error_correction" class="mw-redirect" title="Forward error correction">error correction</a> of transmitted data, and in <a href="/wiki/CD_player" title="CD player">CD players</a>.<a href="#cite_note-FOOTNOTEWelsh1989-75">[59]</a> Another application is <a href="/wiki/Differential_Galois_theory" title="Differential Galois theory">differential Galois theory</a>, which characterizes functions having <a href="/wiki/Antiderivative" title="Antiderivative">antiderivatives</a> of a prescribed form, giving group-theoretic criteria for when solutions of certain <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> are well-behaved.<a href="#cite_note-76">[q]</a> Geometric properties that remain stable under group actions are investigated in <a href="/wiki/Geometric_invariant_theory" title="Geometric invariant theory">(geometric)</a> <a href="/wiki/Invariant_theory" title="Invariant theory">invariant theory</a>.<a href="#cite_note-FOOTNOTEMumfordFogartyKirwan1994-77">[60]</a> <div class="mw-heading mw-heading3"><h3 id="General_linear_group_and_representation_theory">General linear group and representation theory</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=25" title="Edit section: General linear group and representation theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/General_linear_group" title="General linear group">General linear group</a>, <a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a>, and <a href="/wiki/Character_theory" title="Character theory">Character theory</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Matrix_multiplication.svg" class="mw-file-description"><img alt="Two vectors have the same length and span a 90° angle. Furthermore, they are rotated by 90° degrees, then one vector is stretched to twice its length." src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Matrix_multiplication.svg/250px-Matrix_multiplication.svg.png" decoding="async" width="250" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Matrix_multiplication.svg/500px-Matrix_multiplication.svg.png 1.5x" data-file-width="739" data-file-height="274" /></a><figcaption>Two <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vectors</a> (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />⁠-coordinate by factor 2.</figcaption></figure> <a href="/wiki/Matrix_group" class="mw-redirect" title="Matrix group">Matrix groups</a> consist of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> together with <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>. The general linear group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (n,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n,\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c32cfbbae2e2ae75b6647a2bda87942637fe4bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.193ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n,\mathbb {R} )}" /> consists of all <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />⁠-by-⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />⁠ matrices with real entries.<a href="#cite_note-FOOTNOTELay2003-78">[61]</a> Its subgroups are referred to as matrix groups or <a href="/wiki/Linear_group" title="Linear group">linear groups</a>. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the <a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {SO} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa71842f19b6810b4bfa9eb282e92fbf285094e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n)}" />⁠. It describes all possible rotations in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> dimensions. <a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation matrices</a> in this group are used in <a href="/wiki/Computer_graphics" title="Computer graphics">computer graphics</a>.<a href="#cite_note-FOOTNOTEKuipers1999-79">[62]</a> Representation theory is both an application of the group concept and important for a deeper understanding of groups.<a href="#cite_note-FOOTNOTEFultonHarris1991-80">[63]</a><a href="#cite_note-FOOTNOTESerre1977-81">[64]</a> It studies the group by its group actions on other spaces. A broad class of <a href="/wiki/Group_representation" title="Group representation">group representations</a> are linear representations in which the group acts on a vector space, such as the three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}" />⁠. A representation of a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> on an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-<a href="/wiki/Dimension" title="Dimension">dimensional</a> real vector space is simply a group homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e39774b9d55f6696636d0039d2b146f777d7c58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.773ex; height:2.843ex;" alt="{\displaystyle \rho :G\to \mathrm {GL} (n,\mathbb {R} )}" /> from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.<a href="#cite_note-82">[r]</a> A group action gives further means to study the object being acted on.<a href="#cite_note-83">[s]</a> On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and <a href="/wiki/Topological_group" title="Topological group">topological groups</a>, especially (locally) <a href="/wiki/Compact_group" title="Compact group">compact groups</a>.<a href="#cite_note-FOOTNOTEFultonHarris1991-80">[63]</a><a href="#cite_note-FOOTNOTERudin1990-84">[65]</a> <div class="mw-heading mw-heading3"><h3 id="Galois_groups">Galois groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=26" title="Edit section: Galois groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Galois_group" title="Galois group">Galois group</a></div> Galois groups were developed to help solve polynomial equations by capturing their symmetry features.<a href="#cite_note-FOOTNOTERobinson1996viii-85">[66]</a><a href="#cite_note-FOOTNOTEArtin1998-86">[67]</a> For example, the solutions of the <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23e70cfa003f402d108ec04d97983fb62f69536e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.89ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0}" /> are given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9804ca8ce019507e3199ca8fced800fb5b7d7c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.172ex; height:6.176ex;" alt="{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}" /> Each solution can be obtained by replacing the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869e366caf596564de4de06cb0ba124056d4064b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \pm }" /> sign by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" /> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}" />; analogous formulae are known for <a href="/wiki/Cubic_equation" title="Cubic equation">cubic</a> and <a href="/wiki/Quartic_equation" title="Quartic equation">quartic equations</a>, but do not exist in general for <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">degree 5</a> and higher.<a href="#cite_note-FOOTNOTELang2002Chapter_VI_(see_in_particular_p._273_for_concrete_examples)-87">[68]</a> In the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomial equations and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their <a href="/wiki/Solvable_group" title="Solvable group">solvability</a>) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and <a href="/wiki/Nth_root" title="Nth root">roots</a> similar to the formula above.<a href="#cite_note-FOOTNOTELang2002292(Theorem_VI.7.2)-88">[69]</a> Modern <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> generalizes the above type of Galois groups by shifting to field theory and considering <a href="/wiki/Field_extension" title="Field extension">field extensions</a> formed as the <a href="/wiki/Splitting_field" title="Splitting field">splitting field</a> of a polynomial. This theory establishes—via the <a href="/wiki/Fundamental_theorem_of_Galois_theory" title="Fundamental theorem of Galois theory">fundamental theorem of Galois theory</a>—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.<a href="#cite_note-FOOTNOTEStewart2015§12.1-89">[70]</a> <div class="mw-heading mw-heading2"><h2 id="Finite_groups">Finite groups</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=27" title="Edit section: Finite groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_group" title="Finite group">Finite group</a></div> A group is called finite if it has a <a href="/wiki/Finite_set" title="Finite set">finite number of elements</a>. The number of elements is called the <a href="/wiki/Order_of_a_group" class="mw-redirect" title="Order of a group">order</a> of the group.<a href="#cite_note-FOOTNOTEKurzweilStellmacher2004p._3-90">[71]</a> An important class is the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric groups</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {S} _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b24ecef079360cf22a418b01eec68e5a66570d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.984ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{N}}" />⁠, the groups of permutations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> objects. For example, the <a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">symmetric group on 3 letters</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {S} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7739a4f0628592d6e923ae31dac49b5ccd71d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.347ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{3}}" /> is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (<a href="/wiki/Factorial" title="Factorial">factorial</a> of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {S} _{N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b24ecef079360cf22a418b01eec68e5a66570d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.984ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{N}}" /> for a suitable integer ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" />⁠, according to <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a>. Parallel to the group of symmetries of the square above, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {S} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7739a4f0628592d6e923ae31dac49b5ccd71d3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.347ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{3}}" /> can also be interpreted as the group of symmetries of an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>. The order of an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> in a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> is the least positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}=e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fb4e6bbf9772fa3b8abd6bd2fad0e3f97207123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.63ex; height:2.343ex;" alt="{\displaystyle a^{n}=e}" />⁠, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2687184a7698e75db65a25bea7afd207bff3d03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.448ex; height:2.343ex;" alt="{\displaystyle a^{n}}" /> represents <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mo>⋯</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> factors</mtext> </mrow> </mrow> </munder> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1af199ed8f9d2a7838d53345ddba586a83f00829" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:7.023ex; height:5.176ex;" alt="{\displaystyle \underbrace {a\cdots a} _{n{\text{ factors}}},}" /> that is, application of the operation "⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" />⁠" to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> copies of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠. (If "⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" />⁠" represents multiplication, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2687184a7698e75db65a25bea7afd207bff3d03b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.448ex; height:2.343ex;" alt="{\displaystyle a^{n}}" /> corresponds to the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />⁠th power of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" />⁠.) In infinite groups, such an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> may not exist, in which case the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" /> is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element. More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's Theorem</a> states that for a finite group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> the order of any finite subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" /> <a href="/wiki/Divisor" title="Divisor">divides</a> the order of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠. The <a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a> give a partial converse. The dihedral group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> of symmetries of a square is a finite group of order 8. In this group, the order of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}" /> is 4, as is the order of the subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" /> that this element generates. The order of the reflection elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {v} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/691adb625bf4281a64dea47b263114b45b65d78f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.239ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {v} }}" /> etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} _{p}^{\times }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {F} _{p}^{\times }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a46b809ee9c571d5b49222a9d28c76062a3c3414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.931ex; height:3.176ex;" alt="{\displaystyle \mathbb {F} _{p}^{\times }}" /> of multiplication modulo a prime <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> have order ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f356ae51988add41a7da343e6b6d48fa968da162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.262ex; height:2.509ex;" alt="{\displaystyle p-1}" />⁠. <div class="mw-heading mw-heading3"><h3 id="Finite_abelian_groups">Finite abelian groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=28" title="Edit section: Finite abelian groups">edit</a>]</div> Any finite abelian group is isomorphic to a <a href="/wiki/Direct_product" title="Direct product">product</a> of finite cyclic groups; this statement is part of the <a href="/wiki/Fundamental_theorem_of_finitely_generated_abelian_groups" class="mw-redirect" title="Fundamental theorem of finitely generated abelian groups">fundamental theorem of finitely generated abelian groups</a>. Any group of prime order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" /> is isomorphic to the cyclic group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b776b3093bb67b9c48a5a573a7af23fd813bb032" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.479ex; height:2.843ex;" alt="{\displaystyle \mathrm {Z} _{p}}" /> (a consequence of <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a>). Any group of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef685027b97072ee63a8c738f395cd40f63767e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:3.009ex;" alt="{\displaystyle p^{2}}" /> is abelian, isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Z} _{p^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} _{p^{2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70412674c5542a25f69f669c2a701b6080b8d37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.311ex; height:3.009ex;" alt="{\displaystyle \mathrm {Z} _{p^{2}}}" /> or ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d846c95d244a8bf1988a0cb8dcf275af3ece472d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.799ex; height:2.843ex;" alt="{\displaystyle \mathrm {Z} _{p}\times \mathrm {Z} _{p}}" />⁠. But there exist nonabelian groups of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddd72168a6110be2b0bd12486f30b4d40c2d4608" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.313ex; height:3.009ex;" alt="{\displaystyle p^{3}}" />; the dihedral group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {D} _{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109d02b8e5053b48ea9b2bfc1a724ea88a086edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{4}}" /> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e9f8299773e9205d2055998f3c8eb9441877fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{3}}" /> above is an example.<a href="#cite_note-91">[72]</a> <div class="mw-heading mw-heading3"><h3 id="Simple_groups">Simple groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=29" title="Edit section: Simple groups">edit</a>]</div> When a group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> has a normal subgroup <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acdcac635f883f8b4f0a01aa03b16b22f23b124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{1\}}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> itself, questions about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> can sometimes be reduced to questions about <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G/N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab52bff253c690c4e0d473400ab8c365ea019298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.053ex; height:2.843ex;" alt="{\displaystyle G/N}" />⁠. A nontrivial group is called <a href="/wiki/Simple_group" title="Simple group">simple</a> if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the <a href="/wiki/Jordan%E2%80%93H%C3%B6lder_theorem" class="mw-redirect" title="Jordan–Hölder theorem">Jordan–Hölder theorem</a>. <div class="mw-heading mw-heading3"><h3 id="Classification_of_finite_simple_groups">Classification of finite simple groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=30" title="Edit section: Classification of finite simple groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></div> <a href="/wiki/Computer_algebra_system" title="Computer algebra system">Computer algebra systems</a> have been used to <a href="/wiki/List_of_small_groups" title="List of small groups">list all groups of order up to 2000</a>.<a href="#cite_note-92">[t]</a> But <a href="/wiki/Classification_theorems" class="mw-redirect" title="Classification theorems">classifying</a> all finite groups is a problem considered too hard to be solved. The classification of all finite simple groups was a major achievement in contemporary group theory. There are <a href="/wiki/List_of_finite_simple_groups" title="List of finite simple groups">several infinite families</a> of such groups, as well as 26 "<a href="/wiki/Sporadic_groups" class="mw-redirect" title="Sporadic groups">sporadic groups</a>" that do not belong to any of the families. The largest <a href="/wiki/Sporadic_group" title="Sporadic group">sporadic group</a> is called the <a href="/wiki/Monster_group" title="Monster group">monster group</a>. The <a href="/wiki/Monstrous_moonshine" title="Monstrous moonshine">monstrous moonshine</a> conjectures, proved by <a href="/wiki/Richard_Borcherds" title="Richard Borcherds">Richard Borcherds</a>, relate the monster group to certain <a href="/wiki/Modular_function" class="mw-redirect" title="Modular function">modular functions</a>.<a href="#cite_note-FOOTNOTERonan2007-93">[73]</a> The gap between the classification of simple groups and the classification of all groups lies in the <a href="/wiki/Extension_problem" class="mw-redirect" title="Extension problem">extension problem</a>.<a href="#cite_note-94">[74]</a> <div class="mw-heading mw-heading2"><h2 id="Groups_with_additional_structure">Groups with additional structure</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=31" title="Edit section: Groups with additional structure">edit</a>]</div> An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" /> equipped with a binary operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\times G\rightarrow G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\times G\rightarrow G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aca7754938257221721865eb9a5b35ef3e22922" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.935ex; height:2.176ex;" alt="{\displaystyle G\times G\rightarrow G}" /> (the group operation), a <a href="/wiki/Unary_operation" title="Unary operation">unary operation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightarrow G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\rightarrow G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7abf3bf12e9889fa718aae4d71dba45d8120c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:2.176ex;" alt="{\displaystyle G\rightarrow G}" /> (which provides the inverse) and a <a href="/wiki/Nullary_operation" class="mw-redirect" title="Nullary operation">nullary operation</a>, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids <a href="/wiki/Existential_quantifier" class="mw-redirect" title="Existential quantifier">existential quantifiers</a> and is used in computing with groups and for <a href="/wiki/Computer-aided_proof" class="mw-redirect" title="Computer-aided proof">computer-aided proofs</a>. This way of defining groups lends itself to generalizations such as the notion of <a href="/wiki/Group_object" title="Group object">group object</a> in a category. Briefly, this is an object with <a href="/wiki/Morphism" title="Morphism">morphisms</a> that mimic the group axioms.<a href="#cite_note-FOOTNOTEAwodey2010§4.1-95">[75]</a> <div class="mw-heading mw-heading3"><h3 id="Topological_groups">Topological groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=32" title="Edit section: Topological groups">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Circle_as_Lie_group2.svg" class="mw-file-description"><img alt="A part of a circle (highlighted) is projected onto a line." src="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circle_as_Lie_group2.svg/220px-Circle_as_Lie_group2.svg.png" decoding="async" width="220" height="242" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circle_as_Lie_group2.svg/330px-Circle_as_Lie_group2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/de/Circle_as_Lie_group2.svg/440px-Circle_as_Lie_group2.svg.png 2x" data-file-width="377" data-file-height="414" /></a><figcaption>The <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">small piece</a>, such as the red arc in the figure, looks like a part of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> (shown at the bottom).</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Topological_group" title="Topological group">Topological group</a></div> Some <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\cdot h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>⋅</mo> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\cdot h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4ac66a8dc7e57303a6f04d0ee08b5c081ddb3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.134ex; height:2.509ex;" alt="{\displaystyle g\cdot h}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ae77eeb0200a3b0e26ff4b251fb845ca1b385c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:3.009ex;" alt="{\displaystyle g^{-1}}" /> must not vary wildly if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" /> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" /> vary only a little. Such groups are called topological groups, and they are the group objects in the <a href="/wiki/Category_of_topological_spaces" title="Category of topological spaces">category of topological spaces</a>.<a href="#cite_note-FOOTNOTEHusain1966-96">[76]</a> The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological field</a>, such as the field of complex numbers or the field of <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a>. These examples are <a href="/wiki/Locally_compact_topological_group" class="mw-redirect" title="Locally compact topological group">locally compact</a>, so they have <a href="/wiki/Haar_measure" title="Haar measure">Haar measures</a> and can be studied via <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. Other locally compact topological groups include the group of points of an algebraic group over a <a href="/wiki/Local_field" title="Local field">local field</a> or <a href="/wiki/Adele_ring" title="Adele ring">adele ring</a>; these are basic to number theory<a href="#cite_note-FOOTNOTENeukirch1999-97">[77]</a> Galois groups of infinite algebraic field extensions are equipped with the <a href="/wiki/Krull_topology" class="mw-redirect" title="Krull topology">Krull topology</a>, which plays a role in <a href="/wiki/Fundamental_theorem_of_Galois_theory#Infinite_case" title="Fundamental theorem of Galois theory">infinite Galois theory</a>.<a href="#cite_note-FOOTNOTEShatz1972-98">[78]</a> A generalization used in algebraic geometry is the <a href="/wiki/%C3%89tale_fundamental_group" title="Étale fundamental group">étale fundamental group</a>.<a href="#cite_note-FOOTNOTEMilne1980-99">[79]</a> <div class="mw-heading mw-heading3"><h3 id="Lie_groups">Lie groups</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=33" title="Edit section: Lie groups">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lie_group" title="Lie group">Lie group</a></div> A Lie group is a group that also has the structure of a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>; informally, this means that it <a href="/wiki/Diffeomorphism" title="Diffeomorphism">looks locally like</a> a Euclidean space of some fixed dimension.<a href="#cite_note-FOOTNOTEWarner1983-100">[80]</a> Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be <a href="/wiki/Smooth_map" class="mw-redirect" title="Smooth map">smooth</a>. A standard example is the general linear group introduced above: it is an <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subset</a> of the space of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-by-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> matrices, because it is given by the inequality <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)\neq 0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)\neq 0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f9033e8093cb2d8862a7e36d6e47ea669e2329" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.69ex; height:2.843ex;" alt="{\displaystyle \det(A)\neq 0,}" /> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" /> denotes an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" />-by-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /> matrix.<a href="#cite_note-FOOTNOTEBorel1991-101">[81]</a> Lie groups are of fundamental importance in modern physics: <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a> links continuous symmetries to <a href="/wiki/Conserved_quantities" class="mw-redirect" title="Conserved quantities">conserved quantities</a>.<a href="#cite_note-FOOTNOTEGoldstein1980-102">[82]</a> <a href="/wiki/Rotation" title="Rotation">Rotation</a>, as well as translations in <a href="/wiki/Space" title="Space">space</a> and <a href="/wiki/Time" title="Time">time</a>, are basic symmetries of the laws of <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.<a href="#cite_note-103">[u]</a> Another example is the group of <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of <a href="/wiki/Minkowski_space" title="Minkowski space">Minkowski space</a>. The latter serves—in the absence of significant <a href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</a>—as a model of <a href="/wiki/Spacetime" title="Spacetime">spacetime</a> in <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>.<a href="#cite_note-FOOTNOTEWeinberg1972-104">[83]</a> The full symmetry group of Minkowski space, i.e., including translations, is known as the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>. By the above, it plays a pivotal role in special relativity and, by implication, for <a href="/wiki/Quantum_field_theories" class="mw-redirect" title="Quantum field theories">quantum field theories</a>.<a href="#cite_note-FOOTNOTENaber2003-105">[84]</a> <a href="/wiki/Local_symmetry" class="mw-redirect" title="Local symmetry">Symmetries that vary with location</a> are central to the modern description of physical interactions with the help of <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a>. An important example of a gauge theory is the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>, which describes three of the four known <a href="/wiki/Fundamental_force" class="mw-redirect" title="Fundamental force">fundamental forces</a> and classifies all known <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a>.<a href="#cite_note-FOOTNOTEZee2010-106">[85]</a> <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=34" title="Edit section: Generalizations">edit</a>]</div> <table class="floatright" style="font-size:90%"> <caption style="background:#cee0f2; padding:0.2em 0.4em 0.2em; text-align:center; font-size:130%; line-height:1.2em; font-weight: bold">Group-like structures </caption> <tbody><tr> <th> </th> <th><a href="/wiki/Total_function" class="mw-redirect" title="Total function">Total</a> </th> <th><a href="/wiki/Associative_property" title="Associative property">Associative</a> </th> <th><a href="/wiki/Identity_element" title="Identity element">Identity</a> </th> <th><a href="/wiki/Quasigroup" title="Quasigroup"><style data-mw-deduplicate="TemplateStyles:r1038841319">'"`UNIQ--templatestyles-00000345-QINU`"'</style>Divisible</a> </th> <th><a href="/wiki/Commutative_property" title="Commutative property">Commutative</a> </th></tr> <tr> <th><a href="/wiki/Partial_groupoid" title="Partial groupoid">Partial magma</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Semigroupoid" title="Semigroupoid">Semigroupoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Small category</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Groupoid" title="Groupoid">Groupoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Magma_(algebra)" title="Magma (algebra)">magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Unital_magma" class="mw-redirect" title="Unital magma">Unital magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Unital_magma" class="mw-redirect" title="Unital magma">unital magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Loop_(algebra)" class="mw-redirect" title="Loop (algebra)">Loop</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Loop_(algebra)" class="mw-redirect" title="Loop (algebra)">loop</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th>Associative <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative-and-associative <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Monoid" title="Monoid">Monoid</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">Commutative monoid</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a class="mw-selflink selflink">Group</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr></tbody></table> More general structures may be defined by relaxing some of the axioms defining a group.<a href="#cite_note-FOOTNOTEMac_Lane1998-36">[31]</a><a href="#cite_note-FOOTNOTEDeneckeWismath2002-107">[86]</a><a href="#cite_note-FOOTNOTERomanowskaSmith2002-108">[87]</a> The table gives a list of several structures generalizing groups. For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a <a href="/wiki/Monoid" title="Monoid">monoid</a>. The <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" /> (including zero) under addition form a monoid, as do the nonzero integers under multiplication ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98676f71d4f44bfd2294d793f7af37f8e34dd2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}" />⁠. Adjoining inverses of all elements of the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98676f71d4f44bfd2294d793f7af37f8e34dd2a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} \smallsetminus \{0\},\cdot )}" /> produces a group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>∖</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo>⋅</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b1d76e0b728d91db25a20f20823667920f7123e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.626ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Q} \smallsetminus \{0\},\cdot )}" />⁠, and likewise adjoining inverses to any (abelian) monoid ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" />⁠ produces a group known as the <a href="/wiki/Grothendieck_group" title="Grothendieck group">Grothendieck group</a> of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" />⁠. A group can be thought of as a <a href="/wiki/Small_category" class="mw-redirect" title="Small category">small category</a> with one object ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />⁠ in which every morphism is an isomorphism: given such a category, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (x,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (x,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98ade9b2f124f96470824183c830b78adb9a3127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.344ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (x,x)}" /> is a group; conversely, given a group ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" />⁠, one can build a small category with one object ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" />⁠ in which ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (x,x)\simeq G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (x,x)\simeq G}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f381068d269ad3b707f77dc67296298e1a029763" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.269ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (x,x)\simeq G}" />⁠. More generally, a <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> is any small category in which every morphism is an isomorphism. In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06bac4638bb56f14688118ce88c188c7a021eb29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}" />⁠ is defined only when the source of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" />⁠ matches the target of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" />⁠. Groupoids arise in topology (for instance, the <a href="/wiki/Fundamental_groupoid" title="Fundamental groupoid">fundamental groupoid</a>) and in the theory of <a href="/wiki/Stack_(mathematics)" title="Stack (mathematics)">stacks</a>. Finally, it is possible to generalize any of these concepts by replacing the binary operation with an <a href="/wiki/Arity" title="Arity">n-ary</a> operation (i.e., an operation taking n arguments, for some nonnegative integer n). With the proper generalization of the group axioms, this gives a notion of <a href="/wiki/N-ary_group" title="N-ary group">n-ary group</a>.<a href="#cite_note-FOOTNOTEDudek2001-109">[88]</a> <table class="wikitable" style="text-align: center"> <caption>Examples </caption> <tbody><tr> <th>Set </th> <th colspan="2"><a href="/wiki/Natural_number" title="Natural number">Natural numbers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }" />⁠ </th> <th colspan="2"><a href="/wiki/Integer" title="Integer">Integers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" />⁠ </th> <th colspan="4"><a href="/wiki/Rational_number" title="Rational number">Rational numbers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" />⁠ <a href="/wiki/Real_number" title="Real number">Real numbers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" />⁠ <a href="/wiki/Complex_number" title="Complex number">Complex numbers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" />⁠ </th> <th colspan="2"><a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo 3</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} =\{0,1,2\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} =\{0,1,2\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7ff37cd16d007a1ee4daa7130425e7855f225a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.636ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} =\{0,1,2\}}" />⁠ </th></tr> <tr> <th>Operation </th> <td>+ </td> <td>× </td> <td>+ </td> <td>× </td> <td>+ </td> <td>− </td> <td>× </td> <td>÷ </td> <td>+ </td> <td>× </td></tr> <tr> <th><a href="/wiki/Total_function" class="mw-redirect" title="Total function">Total</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr> <tr> <th>Identity </th> <td>0 </td> <td>1 </td> <td>0 </td> <td>1 </td> <td>0 </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>1 </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>0 </td> <td>1 </td></tr> <tr> <th>Inverse </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}" />⁠ </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}" />⁠ </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77da3744bf9edd650bb5dc02004c95129e2bc826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.555ex; height:2.843ex;" alt="{\displaystyle 1/a}" />⁠ (⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}" />⁠) </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td>0, 2, 1, respectively </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No, 1, 2, respectively </td></tr> <tr> <th>Associative </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr> <tr> <th>Commutative </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">No </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Yes </td></tr> <tr> <th>Structure </th> <td><a href="/wiki/Monoid" title="Monoid">monoid</a></td> <td><a href="/wiki/Monoid" title="Monoid">monoid</a> </td> <td><a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> </td> <td><a href="/wiki/Monoid" title="Monoid">monoid</a> </td> <td><a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> </td> <td><a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </td> <td><a href="/wiki/Monoid" title="Monoid">monoid</a> </td> <td><a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> (with 0 removed) </td> <td><a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> </td> <td><a href="/wiki/Monoid" title="Monoid">monoid</a> </td></tr> </tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=35" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul> <ul><li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=36" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-8"><a href="#cite_ref-8">^</a> Some authors include an additional axiom referred to as the closure under the operation "⋅", which means that a ⋅ b is an element of G for every a and b in G. This condition is subsumed by requiring "⋅" to be a binary operation on G. See Lang <a href="#CITEREFLang2002">2002</a>. </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> The <a href="/wiki/MathSciNet" title="MathSciNet">MathSciNet</a> database of mathematics publications lists 1,779 research papers on group theory and its generalizations written in 2020 alone. See <a href="#CITEREFMathSciNet2021">MathSciNet 2021</a>. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> One usually avoids using fraction notation <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠b/a⁠ unless G is abelian, because of the ambiguity of whether it means a−1 ⋅ b or b ⋅ a−1.) </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> See, for example, <a href="#CITEREFLang2002">Lang 2002</a>, <a href="#CITEREFLang2005">Lang 2005</a>, <a href="#CITEREFHerstein1996">Herstein 1996</a> and <a href="#CITEREFHerstein1975">Herstein 1975</a>. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> The word homomorphism derives from <a href="/wiki/Ancient_Greek" title="Ancient Greek">Greek</a> ὁμός—the same and <a href="https://en.wiktionary.org/wiki/%CE%BC%CE%BF%CF%81%CF%86%CE%AE" class="extiw" title="wikt:μορφή">μορφή</a>—structure. See <a href="#CITEREFSchwartzman1994">Schwartzman 1994</a>, p. 108. </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> However, a group is not determined by its lattice of subgroups. See <a href="#CITEREFSuzuki1951">Suzuki 1951</a>. </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> See the <a href="/wiki/Seifert%E2%80%93Van_Kampen_theorem" title="Seifert–Van Kampen theorem">Seifert–Van Kampen theorem</a> for an example. </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> An example is <a href="/wiki/Group_cohomology" title="Group cohomology">group cohomology</a> of a group which equals the <a href="/wiki/Singular_cohomology" class="mw-redirect" title="Singular cohomology">singular cohomology</a> of its <a href="/wiki/Classifying_space" title="Classifying space">classifying space</a>, see <a href="#CITEREFWeibel1994">Weibel 1994</a>, §8.2. </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Elements which do have multiplicative inverses are called <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">units</a>, see <a href="#CITEREFLang2002">Lang 2002</a>, p. 84, §II.1. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> The transition from the integers to the rationals by including fractions is generalized by the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a>. </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> The same is true for any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> F instead of Q. See <a href="#CITEREFLang2005">Lang 2005</a>, p. 86, §III.1. </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> For example, a finite subgroup of the multiplicative group of a field is necessarily cyclic. See <a href="#CITEREFLang2002">Lang 2002</a>, Theorem IV.1.9. The notions of <a href="/wiki/Torsion_(algebra)" title="Torsion (algebra)">torsion</a> of a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> and <a href="/wiki/Simple_algebra" class="mw-redirect" title="Simple algebra">simple algebras</a> are other instances of this principle. </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> The stated property is a possible definition of prime numbers. See <a href="/wiki/Prime_element" title="Prime element">Prime element</a>. </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> For example, the <a href="/wiki/Diffie%E2%80%93Hellman" class="mw-redirect" title="Diffie–Hellman">Diffie–Hellman</a> protocol uses the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a>. See <a href="#CITEREFGollmann2011">Gollmann 2011</a>, §15.3.2. </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> The additive notation for elements of a cyclic group would be t ⋅ a, where t is in Z. </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> More rigorously, every group is the symmetry group of some <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>; see <a href="/wiki/Frucht%27s_theorem" title="Frucht's theorem">Frucht's theorem</a>, <a href="#CITEREFFrucht1939">Frucht 1939</a>. </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> More precisely, the <a href="/wiki/Monodromy" title="Monodromy">monodromy</a> action on the vector space of solutions of the differential equations is considered. See <a href="#CITEREFKuga1993">Kuga 1993</a>, pp. 105–113. </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> This was crucial to the classification of finite simple groups, for example. See <a href="#CITEREFAschbacher2004">Aschbacher 2004</a>. </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> See, for example, <a href="/wiki/Schur%27s_Lemma" class="mw-redirect" title="Schur's Lemma">Schur's Lemma</a> for the impact of a group action on <a href="/wiki/Simple_module" title="Simple module">simple modules</a>. A more involved example is the action of an <a href="/wiki/Absolute_Galois_group" title="Absolute Galois group">absolute Galois group</a> on <a href="/wiki/%C3%89tale_cohomology" title="Étale cohomology">étale cohomology</a>. </li> <li id="cite_note-92"><a href="#cite_ref-92">^</a> <a href="/wiki/Up_to" title="Up to">Up to</a> isomorphism, there are about 49 billion groups of order up to 2000. See <a href="#CITEREFBescheEickO'Brien2001">Besche, Eick & O'Brien 2001</a>. </li> <li id="cite_note-103"><a href="#cite_ref-103">^</a> See <a href="/wiki/Schwarzschild_metric" title="Schwarzschild metric">Schwarzschild metric</a> for an example where symmetry greatly reduces the complexity analysis of physical systems. </li> </ol></div></div> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=37" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEHerstein197526§2-1"><a href="#cite_ref-FOOTNOTEHerstein197526§2_1-0">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 26, §2. </li> <li id="cite_note-FOOTNOTEHall19671§1.1:_"The_idea_of_a_group_is_one_which_pervades_the_whole_of_mathematics_both_[[pure_mathematics|pure]]_and_[[applied_mathematics|applied]]."-2"><a href="#cite_ref-FOOTNOTEHall19671§1.1:_"The_idea_of_a_group_is_one_which_pervades_the_whole_of_mathematics_both_[[pure_mathematics|pure]]_and_[[applied_mathematics|applied]]."_2-0">^</a> <a href="#CITEREFHall1967">Hall 1967</a>, p. 1, §1.1: "The idea of a group is one which pervades the whole of mathematics both <a href="/wiki/Pure_mathematics" title="Pure mathematics">pure</a> and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied</a>." </li> <li id="cite_note-FOOTNOTELang2005360App._2-3"><a href="#cite_ref-FOOTNOTELang2005360App._2_3-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 360, App. 2. </li> <li id="cite_note-FOOTNOTECook200924-4"><a href="#cite_ref-FOOTNOTECook200924_4-0">^</a> <a href="#CITEREFCook2009">Cook 2009</a>, p. 24. </li> <li id="cite_note-FOOTNOTEArtin201840§2.2-5"><a href="#cite_ref-FOOTNOTEArtin201840§2.2_5-0">^</a> <a href="#CITEREFArtin2018">Artin 2018</a>, p. 40, §2.2. </li> <li id="cite_note-FOOTNOTELang2002p._3,_I.§1_and_p._7,_I.§2-6"><a href="#cite_ref-FOOTNOTELang2002p._3,_I.§1_and_p._7,_I.§2_6-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 3, I.§1 and p. 7, I.§2. </li> <li id="cite_note-FOOTNOTELang200516II.§1-7"><a href="#cite_ref-FOOTNOTELang200516II.§1_7-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 16, II.§1. </li> <li id="cite_note-FOOTNOTEHerstein197554§2.6-9"><a href="#cite_ref-FOOTNOTEHerstein197554§2.6_9-0">^</a> <a href="#CITEREFHerstein1975">Herstein 1975</a>, p. 54, §2.6. </li> <li id="cite_note-FOOTNOTEWussing2007-10"><a href="#cite_ref-FOOTNOTEWussing2007_10-0">^</a> <a href="#CITEREFWussing2007">Wussing 2007</a>. </li> <li id="cite_note-FOOTNOTEKleiner1986-11"><a href="#cite_ref-FOOTNOTEKleiner1986_11-0">^</a> <a href="#CITEREFKleiner1986">Kleiner 1986</a>. </li> <li id="cite_note-FOOTNOTESmith1906-12"><a href="#cite_ref-FOOTNOTESmith1906_12-0">^</a> <a href="#CITEREFSmith1906">Smith 1906</a>. </li> <li id="cite_note-FOOTNOTEGalois1908-13"><a href="#cite_ref-FOOTNOTEGalois1908_13-0">^</a> <a href="#CITEREFGalois1908">Galois 1908</a>. </li> <li id="cite_note-FOOTNOTEKleiner1986202-14"><a href="#cite_ref-FOOTNOTEKleiner1986202_14-0">^</a> <a href="#CITEREFKleiner1986">Kleiner 1986</a>, p. 202. </li> <li id="cite_note-FOOTNOTECayley1889-15"><a href="#cite_ref-FOOTNOTECayley1889_15-0">^</a> <a href="#CITEREFCayley1889">Cayley 1889</a>. </li> <li id="cite_note-FOOTNOTEWussing2007§III.2-16"><a href="#cite_ref-FOOTNOTEWussing2007§III.2_16-0">^</a> <a href="#CITEREFWussing2007">Wussing 2007</a>, §III.2. </li> <li id="cite_note-FOOTNOTELie1973-17"><a href="#cite_ref-FOOTNOTELie1973_17-0">^</a> <a href="#CITEREFLie1973">Lie 1973</a>. </li> <li id="cite_note-FOOTNOTEKleiner1986204-18"><a href="#cite_ref-FOOTNOTEKleiner1986204_18-0">^</a> <a href="#CITEREFKleiner1986">Kleiner 1986</a>, p. 204. </li> <li id="cite_note-FOOTNOTEWussing2007§I.3.4-19"><a href="#cite_ref-FOOTNOTEWussing2007§I.3.4_19-0">^</a> <a href="#CITEREFWussing2007">Wussing 2007</a>, §I.3.4. </li> <li id="cite_note-FOOTNOTEJordan1870-20"><a href="#cite_ref-FOOTNOTEJordan1870_20-0">^</a> <a href="#CITEREFJordan1870">Jordan 1870</a>. </li> <li id="cite_note-FOOTNOTEvon_Dyck1882-21"><a href="#cite_ref-FOOTNOTEvon_Dyck1882_21-0">^</a> <a href="#CITEREFvon_Dyck1882">von Dyck 1882</a>. </li> <li id="cite_note-FOOTNOTECurtis2003-22"><a href="#cite_ref-FOOTNOTECurtis2003_22-0">^</a> <a href="#CITEREFCurtis2003">Curtis 2003</a>. </li> <li id="cite_note-FOOTNOTEMackey1976-23"><a href="#cite_ref-FOOTNOTEMackey1976_23-0">^</a> <a href="#CITEREFMackey1976">Mackey 1976</a>. </li> <li id="cite_note-FOOTNOTEBorel2001-24"><a href="#cite_ref-FOOTNOTEBorel2001_24-0">^</a> <a href="#CITEREFBorel2001">Borel 2001</a>. </li> <li id="cite_note-FOOTNOTESolomon2018-25"><a href="#cite_ref-FOOTNOTESolomon2018_25-0">^</a> <a href="#CITEREFSolomon2018">Solomon 2018</a>. </li> <li id="cite_note-FOOTNOTELedermann19534–5§1.2-27"><a href="#cite_ref-FOOTNOTELedermann19534–5§1.2_27-0">^</a> <a href="#CITEREFLedermann1953">Ledermann 1953</a>, pp. 4–5, §1.2. </li> <li id="cite_note-FOOTNOTELedermann19733§I.1-28"><a href="#cite_ref-FOOTNOTELedermann19733§I.1_28-0">^</a> <a href="#CITEREFLedermann1973">Ledermann 1973</a>, p. 3, §I.1. </li> <li id="cite_note-FOOTNOTELang200517§II.1-29">^ <a href="#cite_ref-FOOTNOTELang200517§II.1_29-0">a</a> <a href="#cite_ref-FOOTNOTELang200517§II.1_29-1">b</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 17, §II.1. </li> <li id="cite_note-FOOTNOTEArtin201840-31"><a href="#cite_ref-FOOTNOTEArtin201840_31-0">^</a> <a href="#CITEREFArtin2018">Artin 2018</a>, p. 40. </li> <li id="cite_note-FOOTNOTELang20027§I.2-32">^ <a href="#cite_ref-FOOTNOTELang20027§I.2_32-0">a</a> <a href="#cite_ref-FOOTNOTELang20027§I.2_32-1">b</a> <a href="#cite_ref-FOOTNOTELang20027§I.2_32-2">c</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 7, §I.2. </li> <li id="cite_note-FOOTNOTELang200534§II.3-35"><a href="#cite_ref-FOOTNOTELang200534§II.3_35-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 34, §II.3. </li> <li id="cite_note-FOOTNOTEMac_Lane1998-36">^ <a href="#cite_ref-FOOTNOTEMac_Lane1998_36-0">a</a> <a href="#cite_ref-FOOTNOTEMac_Lane1998_36-1">b</a> <a href="#CITEREFMac_Lane1998">Mac Lane 1998</a>. </li> <li id="cite_note-FOOTNOTELang200519§II.1-37"><a href="#cite_ref-FOOTNOTELang200519§II.1_37-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 19, §II.1. </li> <li id="cite_note-FOOTNOTELedermann197339§II.12-39"><a href="#cite_ref-FOOTNOTELedermann197339§II.12_39-0">^</a> <a href="#CITEREFLedermann1973">Ledermann 1973</a>, p. 39, §II.12. </li> <li id="cite_note-FOOTNOTELang200541§II.4-40"><a href="#cite_ref-FOOTNOTELang200541§II.4_40-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 41, §II.4. </li> <li id="cite_note-FOOTNOTELang200212§I.2-41"><a href="#cite_ref-FOOTNOTELang200212§I.2_41-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 12, §I.2. </li> <li id="cite_note-FOOTNOTELang200545§II.4-42"><a href="#cite_ref-FOOTNOTELang200545§II.4_42-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 45, §II.4. </li> <li id="cite_note-FOOTNOTELang20029§I.2-43"><a href="#cite_ref-FOOTNOTELang20029§I.2_43-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 9, §I.2. </li> <li id="cite_note-FOOTNOTEMagnusKarrassSolitar200456–67§1.6-44"><a href="#cite_ref-FOOTNOTEMagnusKarrassSolitar200456–67§1.6_44-0">^</a> <a href="#CITEREFMagnusKarrassSolitar2004">Magnus, Karrass & Solitar 2004</a>, pp. 56–67, §1.6. </li> <li id="cite_note-FOOTNOTEHatcher200230Chapter_I-45"><a href="#cite_ref-FOOTNOTEHatcher200230Chapter_I_45-0">^</a> <a href="#CITEREFHatcher2002">Hatcher 2002</a>, p. 30, Chapter I. </li> <li id="cite_note-FOOTNOTECoornaertDelzantPapadopoulos1990-48"><a href="#cite_ref-FOOTNOTECoornaertDelzantPapadopoulos1990_48-0">^</a> <a href="#CITEREFCoornaertDelzantPapadopoulos1990">Coornaert, Delzant & Papadopoulos 1990</a>. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> For example, <a href="/wiki/Class_group" class="mw-redirect" title="Class group">class groups</a> and <a href="/wiki/Picard_group" title="Picard group">Picard groups</a>; see <a href="#CITEREFNeukirch1999">Neukirch 1999</a>, in particular §§I.12 and I.13 </li> <li id="cite_note-FOOTNOTESeress1997-50"><a href="#cite_ref-FOOTNOTESeress1997_50-0">^</a> <a href="#CITEREFSeress1997">Seress 1997</a>. </li> <li id="cite_note-FOOTNOTELang2005Chapter_VII-55"><a href="#cite_ref-FOOTNOTELang2005Chapter_VII_55-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, Chapter VII. </li> <li id="cite_note-FOOTNOTERosen200054&nbsp;(Theorem_2.1)-57"><a href="#cite_ref-FOOTNOTERosen200054&nbsp;(Theorem_2.1)_57-0">^</a> <a href="#CITEREFRosen2000">Rosen 2000</a>, p. 54,  (Theorem 2.1). </li> <li id="cite_note-FOOTNOTELang2005292§VIII.1-58"><a href="#cite_ref-FOOTNOTELang2005292§VIII.1_58-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 292, §VIII.1. </li> <li id="cite_note-FOOTNOTELang200522§II.1-60"><a href="#cite_ref-FOOTNOTELang200522§II.1_60-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 22, §II.1. </li> <li id="cite_note-FOOTNOTELang200526§II.2-62"><a href="#cite_ref-FOOTNOTELang200526§II.2_62-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 26, §II.2. </li> <li id="cite_note-FOOTNOTELang200522§II.1_(example_11)-63"><a href="#cite_ref-FOOTNOTELang200522§II.1_(example_11)_63-0">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, p. 22, §II.1 (example 11). </li> <li id="cite_note-FOOTNOTELang200226,_29§I.5-64"><a href="#cite_ref-FOOTNOTELang200226,_29§I.5_64-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, pp. 26, 29, §I.5. </li> <li id="cite_note-FOOTNOTEEllis2019-65">^ <a href="#cite_ref-FOOTNOTEEllis2019_65-0">a</a> <a href="#cite_ref-FOOTNOTEEllis2019_65-1">b</a> <a href="#CITEREFEllis2019">Ellis 2019</a>. </li> <li id="cite_note-FOOTNOTEWeyl1952-66"><a href="#cite_ref-FOOTNOTEWeyl1952_66-0">^</a> <a href="#CITEREFWeyl1952">Weyl 1952</a>. </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> <a href="#CITEREFConwayDelgado_FriedrichsHusonThurston2001">Conway et al. 2001</a>. See also <a href="#CITEREFBishop1993">Bishop 1993</a> </li> <li id="cite_note-FOOTNOTEWeyl1950197–202-69"><a href="#cite_ref-FOOTNOTEWeyl1950197–202_69-0">^</a> <a href="#CITEREFWeyl1950">Weyl 1950</a>, pp. 197–202. </li> <li id="cite_note-FOOTNOTEDove2003-70"><a href="#cite_ref-FOOTNOTEDove2003_70-0">^</a> <a href="#CITEREFDove2003">Dove 2003</a>. </li> <li id="cite_note-FOOTNOTEZee2010228-71"><a href="#cite_ref-FOOTNOTEZee2010228_71-0">^</a> <a href="#CITEREFZee2010">Zee 2010</a>, p. 228. </li> <li id="cite_note-FOOTNOTEChanceyO'Brien202115,_16-72"><a href="#cite_ref-FOOTNOTEChanceyO'Brien202115,_16_72-0">^</a> <a href="#CITEREFChanceyO'Brien2021">Chancey & O'Brien 2021</a>, pp. 15, 16. </li> <li id="cite_note-FOOTNOTESimons2003§4.2.1-73"><a href="#cite_ref-FOOTNOTESimons2003§4.2.1_73-0">^</a> <a href="#CITEREFSimons2003">Simons 2003</a>, §4.2.1. </li> <li id="cite_note-FOOTNOTEElielWilenMander199482-74"><a href="#cite_ref-FOOTNOTEElielWilenMander199482_74-0">^</a> <a href="#CITEREFElielWilenMander1994">Eliel, Wilen & Mander 1994</a>, p. 82. </li> <li id="cite_note-FOOTNOTEWelsh1989-75"><a href="#cite_ref-FOOTNOTEWelsh1989_75-0">^</a> <a href="#CITEREFWelsh1989">Welsh 1989</a>. </li> <li id="cite_note-FOOTNOTEMumfordFogartyKirwan1994-77"><a href="#cite_ref-FOOTNOTEMumfordFogartyKirwan1994_77-0">^</a> <a href="#CITEREFMumfordFogartyKirwan1994">Mumford, Fogarty & Kirwan 1994</a>. </li> <li id="cite_note-FOOTNOTELay2003-78"><a href="#cite_ref-FOOTNOTELay2003_78-0">^</a> <a href="#CITEREFLay2003">Lay 2003</a>. </li> <li id="cite_note-FOOTNOTEKuipers1999-79"><a href="#cite_ref-FOOTNOTEKuipers1999_79-0">^</a> <a href="#CITEREFKuipers1999">Kuipers 1999</a>. </li> <li id="cite_note-FOOTNOTEFultonHarris1991-80">^ <a href="#cite_ref-FOOTNOTEFultonHarris1991_80-0">a</a> <a href="#cite_ref-FOOTNOTEFultonHarris1991_80-1">b</a> <a href="#CITEREFFultonHarris1991">Fulton & Harris 1991</a>. </li> <li id="cite_note-FOOTNOTESerre1977-81"><a href="#cite_ref-FOOTNOTESerre1977_81-0">^</a> <a href="#CITEREFSerre1977">Serre 1977</a>. </li> <li id="cite_note-FOOTNOTERudin1990-84"><a href="#cite_ref-FOOTNOTERudin1990_84-0">^</a> <a href="#CITEREFRudin1990">Rudin 1990</a>. </li> <li id="cite_note-FOOTNOTERobinson1996viii-85"><a href="#cite_ref-FOOTNOTERobinson1996viii_85-0">^</a> <a href="#CITEREFRobinson1996">Robinson 1996</a>, p. viii. </li> <li id="cite_note-FOOTNOTEArtin1998-86"><a href="#cite_ref-FOOTNOTEArtin1998_86-0">^</a> <a href="#CITEREFArtin1998">Artin 1998</a>. </li> <li id="cite_note-FOOTNOTELang2002Chapter_VI_(see_in_particular_p._273_for_concrete_examples)-87"><a href="#cite_ref-FOOTNOTELang2002Chapter_VI_(see_in_particular_p._273_for_concrete_examples)_87-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, Chapter VI (see in particular p. 273 for concrete examples). </li> <li id="cite_note-FOOTNOTELang2002292(Theorem_VI.7.2)-88"><a href="#cite_ref-FOOTNOTELang2002292(Theorem_VI.7.2)_88-0">^</a> <a href="#CITEREFLang2002">Lang 2002</a>, p. 292, (Theorem VI.7.2). </li> <li id="cite_note-FOOTNOTEStewart2015§12.1-89"><a href="#cite_ref-FOOTNOTEStewart2015§12.1_89-0">^</a> <a href="#CITEREFStewart2015">Stewart 2015</a>, §12.1. </li> <li id="cite_note-FOOTNOTEKurzweilStellmacher2004p._3-90"><a href="#cite_ref-FOOTNOTEKurzweilStellmacher2004p._3_90-0">^</a> <a href="#CITEREFKurzweilStellmacher2004">Kurzweil & Stellmacher 2004</a>, p. 3. </li> <li id="cite_note-91"><a href="#cite_ref-91">^</a> <a href="#CITEREFArtin2018">Artin 2018</a>, Proposition 6.4.3. See also <a href="#CITEREFLang2002">Lang 2002</a>, p. 77 for similar results. </li> <li id="cite_note-FOOTNOTERonan2007-93"><a href="#cite_ref-FOOTNOTERonan2007_93-0">^</a> <a href="#CITEREFRonan2007">Ronan 2007</a>. </li> <li id="cite_note-94"><a href="#cite_ref-94">^</a> <a href="#CITEREFAschbacher2004">Aschbacher 2004</a>, p. 737. </li> <li id="cite_note-FOOTNOTEAwodey2010§4.1-95"><a href="#cite_ref-FOOTNOTEAwodey2010§4.1_95-0">^</a> <a href="#CITEREFAwodey2010">Awodey 2010</a>, §4.1. </li> <li id="cite_note-FOOTNOTEHusain1966-96"><a href="#cite_ref-FOOTNOTEHusain1966_96-0">^</a> <a href="#CITEREFHusain1966">Husain 1966</a>. </li> <li id="cite_note-FOOTNOTENeukirch1999-97"><a href="#cite_ref-FOOTNOTENeukirch1999_97-0">^</a> <a href="#CITEREFNeukirch1999">Neukirch 1999</a>. </li> <li id="cite_note-FOOTNOTEShatz1972-98"><a href="#cite_ref-FOOTNOTEShatz1972_98-0">^</a> <a href="#CITEREFShatz1972">Shatz 1972</a>. </li> <li id="cite_note-FOOTNOTEMilne1980-99"><a href="#cite_ref-FOOTNOTEMilne1980_99-0">^</a> <a href="#CITEREFMilne1980">Milne 1980</a>. </li> <li id="cite_note-FOOTNOTEWarner1983-100"><a href="#cite_ref-FOOTNOTEWarner1983_100-0">^</a> <a href="#CITEREFWarner1983">Warner 1983</a>. </li> <li id="cite_note-FOOTNOTEBorel1991-101"><a href="#cite_ref-FOOTNOTEBorel1991_101-0">^</a> <a href="#CITEREFBorel1991">Borel 1991</a>. </li> <li id="cite_note-FOOTNOTEGoldstein1980-102"><a href="#cite_ref-FOOTNOTEGoldstein1980_102-0">^</a> <a href="#CITEREFGoldstein1980">Goldstein 1980</a>. </li> <li id="cite_note-FOOTNOTEWeinberg1972-104"><a href="#cite_ref-FOOTNOTEWeinberg1972_104-0">^</a> <a href="#CITEREFWeinberg1972">Weinberg 1972</a>. </li> <li id="cite_note-FOOTNOTENaber2003-105"><a href="#cite_ref-FOOTNOTENaber2003_105-0">^</a> <a href="#CITEREFNaber2003">Naber 2003</a>. </li> <li id="cite_note-FOOTNOTEZee2010-106"><a href="#cite_ref-FOOTNOTEZee2010_106-0">^</a> <a href="#CITEREFZee2010">Zee 2010</a>. </li> <li id="cite_note-FOOTNOTEDeneckeWismath2002-107"><a href="#cite_ref-FOOTNOTEDeneckeWismath2002_107-0">^</a> <a href="#CITEREFDeneckeWismath2002">Denecke & Wismath 2002</a>. </li> <li id="cite_note-FOOTNOTERomanowskaSmith2002-108"><a href="#cite_ref-FOOTNOTERomanowskaSmith2002_108-0">^</a> <a href="#CITEREFRomanowskaSmith2002">Romanowska & Smith 2002</a>. </li> <li id="cite_note-FOOTNOTEDudek2001-109"><a href="#cite_ref-FOOTNOTEDudek2001_109-0">^</a> <a href="#CITEREFDudek2001">Dudek 2001</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=38" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 30em"> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=39" title="Edit section: General references">edit</a>]</div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFArtin2018" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (2018), Algebra, <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-468960-9" title="Special:BookSources/978-0-13-468960-9"><bdi>978-0-13-468960-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pub=Prentice+Hall&rft.date=2018&rft.isbn=978-0-13-468960-9&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">, Chapter 2 contains an undergraduate-level exposition of the notions covered in this article.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCook2009" class="citation cs2">Cook, Mariana R. (2009), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=06h8NT77OgMC&q=Richard+Ewen+Borcherds&pg=PA24">Mathematicians: An Outer View of the Inner World</a>, Princeton, N.J.: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-13951-7" title="Special:BookSources/978-0-691-13951-7"><bdi>978-0-691-13951-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematicians%3A+An+Outer+View+of+the+Inner+World&rft.place=Princeton%2C+N.J.&rft.pub=Princeton+University+Press&rft.date=2009&rft.isbn=978-0-691-13951-7&rft.aulast=Cook&rft.aufirst=Mariana+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D06h8NT77OgMC%26q%3DRichard%2BEwen%2BBorcherds%26pg%3DPA24&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHall1967" class="citation cs2"><a href="/wiki/George_G._Hall" title="George G. Hall">Hall, G. G.</a> (1967), Applied Group Theory, American Elsevier Publishing Co., Inc., New York, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0219593">0219593</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Group+Theory&rft.pub=American+Elsevier+Publishing+Co.%2C+Inc.%2C+New+York&rft.date=1967&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0219593%23id-name%3DMR&rft.aulast=Hall&rft.aufirst=G.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">, an elementary introduction.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHerstein1996" class="citation cs2"><a href="/wiki/Israel_Nathan_Herstein" title="Israel Nathan Herstein">Herstein, Israel Nathan</a> (1996), Abstract Algebra (3rd ed.), Upper Saddle River, NJ: Prentice Hall Inc., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-374562-7" title="Special:BookSources/978-0-13-374562-7"><bdi>978-0-13-374562-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1375019">1375019</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra&rft.place=Upper+Saddle+River%2C+NJ&rft.edition=3rd&rft.pub=Prentice+Hall+Inc.&rft.date=1996&rft.isbn=978-0-13-374562-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1375019%23id-name%3DMR&rft.aulast=Herstein&rft.aufirst=Israel+Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHerstein1975" class="citation cs2"><a href="/wiki/Israel_Nathan_Herstein" title="Israel Nathan Herstein">Herstein, Israel Nathan</a> (1975), Topics in Algebra (2nd ed.), Lexington, Mass.: Xerox College Publishing, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0356988">0356988</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+Algebra&rft.place=Lexington%2C+Mass.&rft.edition=2nd&rft.pub=Xerox+College+Publishing&rft.date=1975&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0356988%23id-name%3DMR&rft.aulast=Herstein&rft.aufirst=Israel+Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLang2002" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (2002), Algebra, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 211 (Revised third ed.), New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95385-4" title="Special:BookSources/978-0-387-95385-4"><bdi>978-0-387-95385-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1878556">1878556</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=Revised+third&rft.pub=Springer-Verlag&rft.date=2002&rft.isbn=978-0-387-95385-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1878556%23id-name%3DMR&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLang2005" class="citation cs2"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-22025-3" title="Special:BookSources/978-0-387-22025-3"><bdi>978-0-387-22025-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Undergraduate+Algebra&rft.place=Berlin%2C+New+York&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=978-0-387-22025-3&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLedermann1953" class="citation cs2"><a href="/wiki/Walter_Ledermann" title="Walter Ledermann">Ledermann, Walter</a> (1953), Introduction to the Theory of Finite Groups, Oliver and Boyd, Edinburgh and London, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0054593">0054593</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+the+Theory+of+Finite+Groups&rft.pub=Oliver+and+Boyd%2C+Edinburgh+and+London&rft.date=1953&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0054593%23id-name%3DMR&rft.aulast=Ledermann&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLedermann1973" class="citation cs2"><a href="/wiki/Walter_Ledermann" title="Walter Ledermann">Ledermann, Walter</a> (1973), Introduction to Group Theory, New York: Barnes and Noble, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/795613">795613</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Group+Theory&rft.place=New+York&rft.pub=Barnes+and+Noble&rft.date=1973&rft_id=info%3Aoclcnum%2F795613&rft.aulast=Ledermann&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRobinson1996" class="citation cs2"><a href="/wiki/Derek_J._S._Robinson" title="Derek J. S. Robinson">Robinson, Derek John Scott</a> (1996), A Course in the Theory of Groups, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94461-6" title="Special:BookSources/978-0-387-94461-6"><bdi>978-0-387-94461-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+the+Theory+of+Groups&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1996&rft.isbn=978-0-387-94461-6&rft.aulast=Robinson&rft.aufirst=Derek+John+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Special_references">Special references</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=40" title="Edit section: Special references">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFArtin1998" class="citation cs2"><a href="/wiki/Emil_Artin" title="Emil Artin">Artin, Emil</a> (1998), Galois Theory, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-62342-9" title="Special:BookSources/978-0-486-62342-9"><bdi>978-0-486-62342-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1998&rft.isbn=978-0-486-62342-9&rft.aulast=Artin&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAschbacher2004" class="citation cs2"><a href="/wiki/Michael_Aschbacher" title="Michael Aschbacher">Aschbacher, Michael</a> (2004), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200407/fea-aschbacher.pdf">"The status of the classification of the finite simple groups"</a> (PDF), <a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a>, 51 (7): 736–740</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=The+status+of+the+classification+of+the+finite+simple+groups&rft.volume=51&rft.issue=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E736-%3C%2Fspan%3E740&rft.date=2004&rft.aulast=Aschbacher&rft.aufirst=Michael&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200407%2Ffea-aschbacher.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAwodey2010" class="citation cs2">Awodey, Steve (2010), Category Theory, Oxford University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-958736-0" title="Special:BookSources/978-0-19-958736-0"><bdi>978-0-19-958736-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Category+Theory&rft.pub=Oxford+University+Press&rft.date=2010&rft.isbn=978-0-19-958736-0&rft.aulast=Awodey&rft.aufirst=Steve&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBehlerWicklederChristoffers2014" class="citation cs2">Behler, Florian; Wickleder, Mathias S.; Christoffers, Jens (2014), "Biphenyl and bimesityl tetrasulfonic acid – new linker molecules for coordination polymers", Arkivoc, 2015 (2): 64–75, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3998%2Fark.5550190.p008.911">10.3998/ark.5550190.p008.911</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fspo.5550190.p008.911">2027/spo.5550190.p008.911</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Arkivoc&rft.atitle=Biphenyl+and+bimesityl+tetrasulfonic+acid+%E2%80%93+new+linker+molecules+for+coordination+polymers&rft.volume=2015&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E64-%3C%2Fspan%3E75&rft.date=2014&rft_id=info%3Ahdl%2F2027%2Fspo.5550190.p008.911&rft_id=info%3Adoi%2F10.3998%2Fark.5550190.p008.911&rft.aulast=Behler&rft.aufirst=Florian&rft.au=Wickleder%2C+Mathias+S.&rft.au=Christoffers%2C+Jens&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBersuker2006" class="citation cs2">Bersuker, Isaac (2006), <a rel="nofollow" class="external text" href="https://archive.org/details/jahntellereffect0000bers/page/2">The Jahn–Teller Effect</a>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-82212-2" title="Special:BookSources/0-521-82212-2"><bdi>0-521-82212-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Jahn%E2%80%93Teller+Effect&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=0-521-82212-2&rft.aulast=Bersuker&rft.aufirst=Isaac&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fjahntellereffect0000bers%2Fpage%2F2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBescheEickO'Brien2001" class="citation cs2">Besche, Hans Ulrich; Eick, Bettina; O'Brien, E. A. (2001), <a rel="nofollow" class="external text" href="https://www.ams.org/era/2001-07-01/S1079-6762-01-00087-7/home.html">"The groups of order at most 2000"</a>, Electronic Research Announcements of the American Mathematical Society, 7: 1–4, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS1079-6762-01-00087-7">10.1090/S1079-6762-01-00087-7</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1826989">1826989</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Electronic+Research+Announcements+of+the+American+Mathematical+Society&rft.atitle=The+groups+of+order+at+most+2000&rft.volume=7&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E4&rft.date=2001&rft_id=info%3Adoi%2F10.1090%2FS1079-6762-01-00087-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1826989%23id-name%3DMR&rft.aulast=Besche&rft.aufirst=Hans+Ulrich&rft.au=Eick%2C+Bettina&rft.au=O%27Brien%2C+E.+A.&rft_id=https%3A%2F%2Fwww.ams.org%2Fera%2F2001-07-01%2FS1079-6762-01-00087-7%2Fhome.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBishop1993" class="citation cs2">Bishop, David H. L. (1993), Group Theory and Chemistry, New York: Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-67355-4" title="Special:BookSources/978-0-486-67355-4"><bdi>978-0-486-67355-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory+and+Chemistry&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1993&rft.isbn=978-0-486-67355-4&rft.aulast=Bishop&rft.aufirst=David+H.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBorel1991" class="citation cs2"><a href="/wiki/Armand_Borel" title="Armand Borel">Borel, Armand</a> (1991), Linear Algebraic Groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97370-8" title="Special:BookSources/978-0-387-97370-8"><bdi>978-0-387-97370-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1102012">1102012</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebraic+Groups&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1991&rft.isbn=978-0-387-97370-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1102012%23id-name%3DMR&rft.aulast=Borel&rft.aufirst=Armand&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCarter1989" class="citation cs2"><a href="/wiki/Roger_Carter_(mathematician)" title="Roger Carter (mathematician)">Carter, Roger W.</a> (1989), Simple Groups of Lie Type, New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-50683-6" title="Special:BookSources/978-0-471-50683-6"><bdi>978-0-471-50683-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Simple+Groups+of+Lie+Type&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=978-0-471-50683-6&rft.aulast=Carter&rft.aufirst=Roger+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChanceyO'Brien2021" class="citation cs2">Chancey, C. C.; O'Brien, M. C. M. (2021), The Jahn–Teller Effect in C60 and Other Icosahedral Complexes, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-22534-0" title="Special:BookSources/978-0-691-22534-0"><bdi>978-0-691-22534-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Jahn%E2%80%93Teller+Effect+in+C60+and+Other+Icosahedral+Complexes&rft.pub=Princeton+University+Press&rft.date=2021&rft.isbn=978-0-691-22534-0&rft.aulast=Chancey&rft.aufirst=C.+C.&rft.au=O%27Brien%2C+M.+C.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFConwayDelgado_FriedrichsHusonThurston2001" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John Horton</a>; Delgado Friedrichs, Olaf; Huson, Daniel H.; <a href="/wiki/William_Thurston" title="William Thurston">Thurston, William P.</a> (2001), "On three-dimensional space groups", Beiträge zur Algebra und Geometrie, 42 (2): 475–507, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.MG/9911185">math.MG/9911185</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1865535">1865535</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Beitr%C3%A4ge+zur+Algebra+und+Geometrie&rft.atitle=On+three-dimensional+space+groups&rft.volume=42&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E475-%3C%2Fspan%3E507&rft.date=2001&rft_id=info%3Aarxiv%2Fmath.MG%2F9911185&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1865535%23id-name%3DMR&rft.aulast=Conway&rft.aufirst=John+Horton&rft.au=Delgado+Friedrichs%2C+Olaf&rft.au=Huson%2C+Daniel+H.&rft.au=Thurston%2C+William+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCoornaertDelzantPapadopoulos1990" class="citation cs2 cs1-prop-foreign-lang-source">Coornaert, M.; Delzant, T.; Papadopoulos, A. (1990), Géométrie et théorie des groupes [Geometry and Group Theory], Lecture Notes in Mathematics (in French), vol. 1441, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-52977-4" title="Special:BookSources/978-3-540-52977-4"><bdi>978-3-540-52977-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1075994">1075994</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=G%C3%A9om%C3%A9trie+et+th%C3%A9orie+des+groupes+%5BGeometry+and+Group+Theory%5D&rft.place=Berlin%2C+New+York&rft.series=Lecture+Notes+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1990&rft.isbn=978-3-540-52977-4&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1075994%23id-name%3DMR&rft.aulast=Coornaert&rft.aufirst=M.&rft.au=Delzant%2C+T.&rft.au=Papadopoulos%2C+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDeneckeWismath2002" class="citation cs2">Denecke, Klaus; Wismath, Shelly L. (2002), Universal Algebra and Applications in Theoretical Computer Science, London: <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-254-1" title="Special:BookSources/978-1-58488-254-1"><bdi>978-1-58488-254-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Universal+Algebra+and+Applications+in+Theoretical+Computer+Science&rft.place=London&rft.pub=CRC+Press&rft.date=2002&rft.isbn=978-1-58488-254-1&rft.aulast=Denecke&rft.aufirst=Klaus&rft.au=Wismath%2C+Shelly+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDove2003" class="citation cs2">Dove, Martin T (2003), Structure and Dynamics: An Atomic View of Materials, Oxford University Press, p. 265, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-850678-3" title="Special:BookSources/0-19-850678-3"><bdi>0-19-850678-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Structure+and+Dynamics%3A+An+Atomic+View+of+Materials&rft.pages=265&rft.pub=Oxford+University+Press&rft.date=2003&rft.isbn=0-19-850678-3&rft.aulast=Dove&rft.aufirst=Martin+T&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDudek2001" class="citation cs2">Dudek, Wiesław A. (2001), <a rel="nofollow" class="external text" href="https://ibn.idsi.md/sites/default/files/imag_file/15-36_On%20some%20old%20and%20new%20problems%20in%20n-ary%20groups.pdf">"On some old and new problems in n-ary groups"</a> (PDF), Quasigroups and Related Systems, 8: 15–36, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1876783">1876783</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quasigroups+and+Related+Systems&rft.atitle=On+some+old+and+new+problems+in+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3En%3C%2Fspan%3E-ary+groups&rft.volume=8&rft.pages=%3Cspan+class%3D%22nowrap%22%3E15-%3C%2Fspan%3E36&rft.date=2001&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1876783%23id-name%3DMR&rft.aulast=Dudek&rft.aufirst=Wies%C5%82aw+A.&rft_id=https%3A%2F%2Fibn.idsi.md%2Fsites%2Fdefault%2Ffiles%2Fimag_file%2F15-36_On%2520some%2520old%2520and%2520new%2520problems%2520in%2520n-ary%2520groups.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFElielWilenMander1994" class="citation cs2">Eliel, Ernest; Wilen, Samuel; Mander, Lewis (1994), Stereochemistry of Organic Compounds, Wiley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-01670-0" title="Special:BookSources/978-0-471-01670-0"><bdi>978-0-471-01670-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stereochemistry+of+Organic+Compounds&rft.pub=Wiley&rft.date=1994&rft.isbn=978-0-471-01670-0&rft.aulast=Eliel&rft.aufirst=Ernest&rft.au=Wilen%2C+Samuel&rft.au=Mander%2C+Lewis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFEllis2019" class="citation cs2">Ellis, Graham (2019), "6.4 Triangle groups", An Invitation to Computational Homotopy, Oxford University Press, pp. 441–444, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Foso%2F9780198832973.001.0001">10.1093/oso/9780198832973.001.0001</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-883298-0" title="Special:BookSources/978-0-19-883298-0"><bdi>978-0-19-883298-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3971587">3971587</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6.4+Triangle+groups&rft.btitle=An+Invitation+to+Computational+Homotopy&rft.pages=%3Cspan+class%3D%22nowrap%22%3E441-%3C%2Fspan%3E444&rft.pub=Oxford+University+Press&rft.date=2019&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3971587%23id-name%3DMR&rft_id=info%3Adoi%2F10.1093%2Foso%2F9780198832973.001.0001&rft.isbn=978-0-19-883298-0&rft.aulast=Ellis&rft.aufirst=Graham&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFrucht1939" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Robert_Frucht" title="Robert Frucht">Frucht, R.</a> (1939), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081201083831/http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0">"Herstellung von Graphen mit vorgegebener abstrakter Gruppe [Construction of graphs with prescribed group]"</a>, Compositio Mathematica (in German), 6: 239–50, archived from <a rel="nofollow" class="external text" href="http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0">the original</a> on 2008-12-01</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Compositio+Mathematica&rft.atitle=Herstellung+von+Graphen+mit+vorgegebener+abstrakter+Gruppe+%5BConstruction+of+graphs+with+prescribed+group%5D&rft.volume=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E239-%3C%2Fspan%3E50&rft.date=1939&rft.aulast=Frucht&rft.aufirst=R.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fnumdam-bin%2Ffitem%3Fid%3DCM_1939__6__239_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFultonHarris1991" class="citation cs2"><a href="/wiki/William_Fulton_(mathematician)" title="William Fulton (mathematician)">Fulton, William</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (1991), Representation Theory: A First Course, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, Readings in Mathematics, vol. 129, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97495-8" title="Special:BookSources/978-0-387-97495-8"><bdi>978-0-387-97495-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1153249">1153249</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representation+Theory%3A+A+First+Course&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics%2C+Readings+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1991&rft.isbn=978-0-387-97495-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1153249%23id-name%3DMR&rft.aulast=Fulton&rft.aufirst=William&rft.au=Harris%2C+Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGoldstein1980" class="citation cs2"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a> (1980), <a href="/wiki/Classical_Mechanics_(Goldstein)" title="Classical Mechanics (Goldstein)">Classical Mechanics</a> (2nd ed.), Reading, MA: Addison-Wesley Publishing, pp. 588–596, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-02918-9" title="Special:BookSources/0-201-02918-9"><bdi>0-201-02918-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.place=Reading%2C+MA&rft.pages=%3Cspan+class%3D%22nowrap%22%3E588-%3C%2Fspan%3E596&rft.edition=2nd&rft.pub=Addison-Wesley+Publishing&rft.date=1980&rft.isbn=0-201-02918-9&rft.aulast=Goldstein&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGollmann2011" class="citation cs2">Gollmann, Dieter (2011), Computer Security (2nd ed.), West Sussex, England: John Wiley & Sons, Ltd., <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-74115-3" title="Special:BookSources/978-0-470-74115-3"><bdi>978-0-470-74115-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computer+Security&rft.place=West+Sussex%2C+England&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons%2C+Ltd.&rft.date=2011&rft.isbn=978-0-470-74115-3&rft.aulast=Gollmann&rft.aufirst=Dieter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHatcher2002" class="citation cs2"><a href="/wiki/Allen_Hatcher" title="Allen Hatcher">Hatcher, Allen</a> (2002), <a rel="nofollow" class="external text" href="http://pi.math.cornell.edu/~hatcher/AT/ATpage.html">Algebraic Topology</a>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-79540-1" title="Special:BookSources/978-0-521-79540-1"><bdi>978-0-521-79540-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Topology&rft.pub=Cambridge+University+Press&rft.date=2002&rft.isbn=978-0-521-79540-1&rft.aulast=Hatcher&rft.aufirst=Allen&rft_id=http%3A%2F%2Fpi.math.cornell.edu%2F~hatcher%2FAT%2FATpage.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHusain1966" class="citation cs2">Husain, Taqdir (1966), Introduction to Topological Groups, Philadelphia: W.B. Saunders Company, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-89874-193-3" title="Special:BookSources/978-0-89874-193-3"><bdi>978-0-89874-193-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Topological+Groups&rft.place=Philadelphia&rft.pub=W.B.+Saunders+Company&rft.date=1966&rft.isbn=978-0-89874-193-3&rft.aulast=Husain&rft.aufirst=Taqdir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJahnTeller1937" class="citation cs2"><a href="/wiki/Hermann_Arthur_Jahn" title="Hermann Arthur Jahn">Jahn, H.</a>; <a href="/wiki/Edward_Teller" title="Edward Teller">Teller, E.</a> (1937), "Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 161 (905): 220–235, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1937RSPSA.161..220J">1937RSPSA.161..220J</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1937.0142">10.1098/rspa.1937.0142</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=Stability+of+polyatomic+molecules+in+degenerate+electronic+states.+I.+Orbital+degeneracy&rft.volume=161&rft.issue=905&rft.pages=%3Cspan+class%3D%22nowrap%22%3E220-%3C%2Fspan%3E235&rft.date=1937&rft_id=info%3Adoi%2F10.1098%2Frspa.1937.0142&rft_id=info%3Abibcode%2F1937RSPSA.161..220J&rft.aulast=Jahn&rft.aufirst=H.&rft.au=Teller%2C+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKuipers1999" class="citation cs2">Kuipers, Jack B. (1999), Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999qrsp.book.....K">1999qrsp.book.....K</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-05872-6" title="Special:BookSources/978-0-691-05872-6"><bdi>978-0-691-05872-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1670862">1670862</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quaternions+and+Rotation+Sequences%3A+A+Primer+with+Applications+to+Orbits%2C+Aerospace%2C+and+Virtual+Reality&rft.pub=Princeton+University+Press&rft.date=1999&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1670862%23id-name%3DMR&rft_id=info%3Abibcode%2F1999qrsp.book.....K&rft.isbn=978-0-691-05872-6&rft.aulast=Kuipers&rft.aufirst=Jack+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKuga1993" class="citation cs2"><a href="/wiki/Michio_Kuga" title="Michio Kuga">Kuga, Michio</a> (1993), <a rel="nofollow" class="external text" href="https://archive.org/details/galoisdreamgroup0000kuga">Galois' Dream: Group Theory and Differential Equations</a>, Boston, MA: Birkhäuser Boston, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-3688-3" title="Special:BookSources/978-0-8176-3688-3"><bdi>978-0-8176-3688-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1199112">1199112</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois%27+Dream%3A+Group+Theory+and+Differential+Equations&rft.place=Boston%2C+MA&rft.pub=Birkh%C3%A4user+Boston&rft.date=1993&rft.isbn=978-0-8176-3688-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1199112%23id-name%3DMR&rft.aulast=Kuga&rft.aufirst=Michio&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgaloisdreamgroup0000kuga&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKurzweilStellmacher2004" class="citation cs2">Kurzweil, Hans; Stellmacher, Bernd (2004), The Theory of Finite Groups, Universitext, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-40510-0" title="Special:BookSources/978-0-387-40510-0"><bdi>978-0-387-40510-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2014408">2014408</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Finite+Groups&rft.place=Berlin%2C+New+York&rft.series=Universitext&rft.pub=Springer-Verlag&rft.date=2004&rft.isbn=978-0-387-40510-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2014408%23id-name%3DMR&rft.aulast=Kurzweil&rft.aufirst=Hans&rft.au=Stellmacher%2C+Bernd&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLay2003" class="citation cs2">Lay, David (2003), Linear Algebra and Its Applications, <a href="/wiki/Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-70970-4" title="Special:BookSources/978-0-201-70970-4"><bdi>978-0-201-70970-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra+and+Its+Applications&rft.pub=Addison-Wesley&rft.date=2003&rft.isbn=978-0-201-70970-4&rft.aulast=Lay&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMac_Lane1998" class="citation cs2"><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1998), <a href="/wiki/Categories_for_the_Working_Mathematician" title="Categories for the Working Mathematician">Categories for the Working Mathematician</a> (2nd ed.), Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98403-2" title="Special:BookSources/978-0-387-98403-2"><bdi>978-0-387-98403-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Categories+for+the+Working+Mathematician&rft.place=Berlin%2C+New+York&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1998&rft.isbn=978-0-387-98403-2&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMagnusKarrassSolitar2004" class="citation cs2"><a href="/wiki/Wilhelm_Magnus" title="Wilhelm Magnus">Magnus, Wilhelm</a>; Karrass, Abraham; Solitar, Donald (2004) [1966], <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1LW4s1RDRHQC&pg=PR2">Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relations</a>, Courier, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43830-6" title="Special:BookSources/978-0-486-43830-6"><bdi>978-0-486-43830-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+Group+Theory%3A+Presentations+of+Groups+in+Terms+of+Generators+and+Relations&rft.pub=Courier&rft.date=2004&rft.isbn=978-0-486-43830-6&rft.aulast=Magnus&rft.aufirst=Wilhelm&rft.au=Karrass%2C+Abraham&rft.au=Solitar%2C+Donald&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1LW4s1RDRHQC%26pg%3DPR2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMathSciNet2021" class="citation cs2">MathSciNet (2021), <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=20&co6=AND&pg7=ALLF&s7=&co7=AND&dr=pubyear&yrop=eq&arg3=2020&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&review_format=html&Submit=Suche">List of papers reviewed on MathSciNet on "Group theory and its generalizations" (MSC code 20), published in 2020</a>, retrieved 14 May 2021</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=List+of+papers+reviewed+on+MathSciNet+on+%22Group+theory+and+its+generalizations%22+%28MSC+code+20%29%2C+published+in+2020&rft.date=2021&rft.au=MathSciNet&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet%2Fsearch%2Fpublications.html%3Fpg4%3DAUCN%26s4%3D%26co4%3DAND%26pg5%3DTI%26s5%3D%26co5%3DAND%26pg6%3DPC%26s6%3D20%26co6%3DAND%26pg7%3DALLF%26s7%3D%26co7%3DAND%26dr%3Dpubyear%26yrop%3Deq%26arg3%3D2020%26yearRangeFirst%3D%26yearRangeSecond%3D%26pg8%3DET%26s8%3DAll%26review_format%3Dhtml%26Submit%3DSuche&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMichler2006" class="citation cs2">Michler, Gerhard (2006), Theory of Finite Simple Groups, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86625-5" title="Special:BookSources/978-0-521-86625-5"><bdi>978-0-521-86625-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Finite+Simple+Groups&rft.pub=Cambridge+University+Press&rft.date=2006&rft.isbn=978-0-521-86625-5&rft.aulast=Michler&rft.aufirst=Gerhard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMilne1980" class="citation cs2">Milne, James S. (1980), <a rel="nofollow" class="external text" href="https://archive.org/details/etalecohomology00miln">Étale Cohomology</a>, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08238-7" title="Special:BookSources/978-0-691-08238-7"><bdi>978-0-691-08238-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89tale+Cohomology&rft.pub=Princeton+University+Press&rft.date=1980&rft.isbn=978-0-691-08238-7&rft.aulast=Milne&rft.aufirst=James+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fetalecohomology00miln&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMumfordFogartyKirwan1994" class="citation cs2"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a>; Fogarty, J.; Kirwan, F. (1994), Geometric Invariant Theory, vol. 34 (3rd ed.), Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-56963-3" title="Special:BookSources/978-3-540-56963-3"><bdi>978-3-540-56963-3</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1304906">1304906</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+Invariant+Theory&rft.place=Berlin%2C+New+York&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-3-540-56963-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1304906%23id-name%3DMR&rft.aulast=Mumford&rft.aufirst=David&rft.au=Fogarty%2C+J.&rft.au=Kirwan%2C+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNaber2003" class="citation cs2">Naber, Gregory L. (2003), The Geometry of Minkowski Spacetime, New York: Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43235-9" title="Special:BookSources/978-0-486-43235-9"><bdi>978-0-486-43235-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2044239">2044239</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Minkowski+Spacetime&rft.place=New+York&rft.pub=Dover+Publications&rft.date=2003&rft.isbn=978-0-486-43235-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2044239%23id-name%3DMR&rft.aulast=Naber&rft.aufirst=Gregory+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNeukirch1999" class="citation cs2"><a href="/wiki/J%C3%BCrgen_Neukirch" title="Jürgen Neukirch">Neukirch, Jürgen</a> (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, vol. 322, Berlin: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-65399-8" title="Special:BookSources/978-3-540-65399-8"><bdi>978-3-540-65399-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1697859">1697859</a>, <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0956.11021">0956.11021</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Number+Theory&rft.place=Berlin&rft.series=%3Cspan+title%3D%22German-language+text%22%3E%3Ci+lang%3D%22de%22%3EGrundlehren+der+mathematischen+Wissenschaften%3C%2Fi%3E%3C%2Fspan%3ECategory%3AArticles+containing+German-language+text&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0956.11021%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1697859%23id-name%3DMR&rft.isbn=978-3-540-65399-8&rft.aulast=Neukirch&rft.aufirst=J%C3%BCrgen&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRomanowskaSmith2002" class="citation cs2"><a href="/wiki/Anna_Romanowska" title="Anna Romanowska">Romanowska, A. B.</a>; Smith, J. D. H. (2002), Modes, <a href="/wiki/World_Scientific" title="World Scientific">World Scientific</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-4942-7" title="Special:BookSources/978-981-02-4942-7"><bdi>978-981-02-4942-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modes&rft.pub=World+Scientific&rft.date=2002&rft.isbn=978-981-02-4942-7&rft.aulast=Romanowska&rft.aufirst=A.+B.&rft.au=Smith%2C+J.+D.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRonan2007" class="citation cs2"><a href="/wiki/Mark_Ronan" title="Mark Ronan">Ronan, Mark</a> (2007), Symmetry and the Monster: The Story of One of the Greatest Quests of Mathematics, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-280723-6" title="Special:BookSources/978-0-19-280723-6"><bdi>978-0-19-280723-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry+and+the+Monster%3A+The+Story+of+One+of+the+Greatest+Quests+of+Mathematics&rft.pub=Oxford+University+Press&rft.date=2007&rft.isbn=978-0-19-280723-6&rft.aulast=Ronan&rft.aufirst=Mark&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRosen2000" class="citation cs2">Rosen, Kenneth H. (2000), Elementary Number Theory and its Applications (4th ed.), Addison-Wesley, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-201-87073-2" title="Special:BookSources/978-0-201-87073-2"><bdi>978-0-201-87073-2</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1739433">1739433</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Number+Theory+and+its+Applications&rft.edition=4th&rft.pub=Addison-Wesley&rft.date=2000&rft.isbn=978-0-201-87073-2&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1739433%23id-name%3DMR&rft.aulast=Rosen&rft.aufirst=Kenneth+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRudin1990" class="citation cs2"><a href="/wiki/Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1990), Fourier Analysis on Groups, Wiley Classics, Wiley-Blackwell, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-52364-X" title="Special:BookSources/0-471-52364-X"><bdi>0-471-52364-X</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fourier+Analysis+on+Groups&rft.series=Wiley+Classics&rft.pub=Wiley-Blackwell&rft.date=1990&rft.isbn=0-471-52364-X&rft.aulast=Rudin&rft.aufirst=Walter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSeress1997" class="citation cs2">Seress, Ákos (1997), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/199706/seress.pdf">"An Introduction to Computational Group Theory"</a> (PDF), Notices of the American Mathematical Society, 44 (6): 671–679, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1452069">1452069</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=An+Introduction+to+Computational+Group+Theory&rft.volume=44&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E671-%3C%2Fspan%3E679&rft.date=1997&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1452069%23id-name%3DMR&rft.aulast=Seress&rft.aufirst=%C3%81kos&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F199706%2Fseress.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSerre1977" class="citation cs2"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1977), <a rel="nofollow" class="external text" href="https://archive.org/details/linearrepresenta1977serr">Linear Representations of Finite Groups</a>, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90190-9" title="Special:BookSources/978-0-387-90190-9"><bdi>978-0-387-90190-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0450380">0450380</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Representations+of+Finite+Groups&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=978-0-387-90190-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0450380%23id-name%3DMR&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flinearrepresenta1977serr&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSchwartzman1994" class="citation cs2">Schwartzman, Steven (1994), The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English, Mathematical Association of America, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-511-9" title="Special:BookSources/978-0-88385-511-9"><bdi>978-0-88385-511-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Words+of+Mathematics%3A+An+Etymological+Dictionary+of+Mathematical+Terms+Used+in+English&rft.pub=Mathematical+Association+of+America&rft.date=1994&rft.isbn=978-0-88385-511-9&rft.aulast=Schwartzman&rft.aufirst=Steven&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFShatz1972" class="citation cs2">Shatz, Stephen S. (1972), Profinite Groups, Arithmetic, and Geometry, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08017-8" title="Special:BookSources/978-0-691-08017-8"><bdi>978-0-691-08017-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0347778">0347778</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Profinite+Groups%2C+Arithmetic%2C+and+Geometry&rft.pub=Princeton+University+Press&rft.date=1972&rft.isbn=978-0-691-08017-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0347778%23id-name%3DMR&rft.aulast=Shatz&rft.aufirst=Stephen+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSimons2003" class="citation cs2">Simons, Jack (2003), An Introduction to Theoretical Chemistry, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-53047-7" title="Special:BookSources/978-0-521-53047-7"><bdi>978-0-521-53047-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Theoretical+Chemistry&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-53047-7&rft.aulast=Simons&rft.aufirst=Jack&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSolomon2018" class="citation cs2">Solomon, Ronald (2018), "The classification of finite simple groups: A progress report", Notices of the AMS, 65 (6): 1, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fnoti1689">10.1090/noti1689</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+AMS&rft.atitle=The+classification+of+finite+simple+groups%3A+A+progress+report&rft.volume=65&rft.issue=6&rft.pages=1&rft.date=2018&rft_id=info%3Adoi%2F10.1090%2Fnoti1689&rft.aulast=Solomon&rft.aufirst=Ronald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStewart2015" class="citation cs2"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Stewart, Ian</a> (2015), Galois Theory (4th ed.), CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4822-4582-0" title="Special:BookSources/978-1-4822-4582-0"><bdi>978-1-4822-4582-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory&rft.edition=4th&rft.pub=CRC+Press&rft.date=2015&rft.isbn=978-1-4822-4582-0&rft.aulast=Stewart&rft.aufirst=Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSuzuki1951" class="citation cs2"><a href="/wiki/Michio_Suzuki_(mathematician)" title="Michio Suzuki (mathematician)">Suzuki, Michio</a> (1951), "On the lattice of subgroups of finite groups", <a href="/wiki/Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>, 70 (2): 345–371, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990375">10.2307/1990375</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990375">1990375</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=On+the+lattice+of+subgroups+of+finite+groups&rft.volume=70&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E345-%3C%2Fspan%3E371&rft.date=1951&rft_id=info%3Adoi%2F10.2307%2F1990375&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1990375%23id-name%3DJSTOR&rft.aulast=Suzuki&rft.aufirst=Michio&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWarner1983" class="citation cs2">Warner, Frank (1983), Foundations of Differentiable Manifolds and Lie Groups, Berlin, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90894-6" title="Special:BookSources/978-0-387-90894-6"><bdi>978-0-387-90894-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Differentiable+Manifolds+and+Lie+Groups&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=1983&rft.isbn=978-0-387-90894-6&rft.aulast=Warner&rft.aufirst=Frank&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeibel1994" class="citation book cs2"><a href="/wiki/Charles_Weibel" title="Charles Weibel">Weibel, Charles A.</a> (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55987-4" title="Special:BookSources/978-0-521-55987-4"><bdi>978-0-521-55987-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1269324">1269324</a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36131259">36131259</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+homological+algebra&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=info%3Aoclcnum%2F36131259&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1269324%23id-name%3DMR&rft.isbn=978-0-521-55987-4&rft.aulast=Weibel&rft.aufirst=Charles+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeinberg1972" class="citation cs2"><a href="/wiki/Steven_Weinberg" title="Steven Weinberg">Weinberg, Steven</a> (1972), <a rel="nofollow" class="external text" href="https://archive.org/details/gravitationcosmo00stev_0">Gravitation and Cosmology</a>, New York: John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-92567-5" title="Special:BookSources/0-471-92567-5"><bdi>0-471-92567-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation+and+Cosmology&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1972&rft.isbn=0-471-92567-5&rft.aulast=Weinberg&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgravitationcosmo00stev_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWelsh1989" class="citation cs2"><a href="/wiki/Dominic_Welsh" title="Dominic Welsh">Welsh, Dominic</a> (1989), Codes and Cryptography, Oxford: Clarendon Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-853287-3" title="Special:BookSources/978-0-19-853287-3"><bdi>978-0-19-853287-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Codes+and+Cryptography&rft.place=Oxford&rft.pub=Clarendon+Press&rft.date=1989&rft.isbn=978-0-19-853287-3&rft.aulast=Welsh&rft.aufirst=Dominic&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeyl1952" class="citation cs2"><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl, Hermann</a> (1952), Symmetry, Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-02374-8" title="Special:BookSources/978-0-691-02374-8"><bdi>978-0-691-02374-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Symmetry&rft.pub=Princeton+University+Press&rft.date=1952&rft.isbn=978-0-691-02374-8&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZee2010" class="citation cs2"><a href="/wiki/Anthony_Zee" title="Anthony Zee">Zee, A.</a> (2010), <a href="/wiki/Quantum_Field_Theory_in_a_Nutshell" title="Quantum Field Theory in a Nutshell">Quantum Field Theory in a Nutshell</a> (second ed.), Princeton, N.J.: Princeton University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-14034-6" title="Special:BookSources/978-0-691-14034-6"><bdi>978-0-691-14034-6</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/768477138">768477138</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Field+Theory+in+a+Nutshell&rft.place=Princeton%2C+N.J.&rft.edition=second&rft.pub=Princeton+University+Press&rft.date=2010&rft_id=info%3Aoclcnum%2F768477138&rft.isbn=978-0-691-14034-6&rft.aulast=Zee&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=41" title="Edit section: Historical references">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_publications_in_mathematics#Group_theory" title="List of publications in mathematics">Historically important publications in group theory</a></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBorel2001" class="citation cs2"><a href="/wiki/Armand_Borel" title="Armand Borel">Borel, Armand</a> (2001), Essays in the History of Lie Groups and Algebraic Groups, Providence, R.I.: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-0288-5" title="Special:BookSources/978-0-8218-0288-5"><bdi>978-0-8218-0288-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essays+in+the+History+of+Lie+Groups+and+Algebraic+Groups&rft.place=Providence%2C+R.I.&rft.pub=American+Mathematical+Society&rft.date=2001&rft.isbn=978-0-8218-0288-5&rft.aulast=Borel&rft.aufirst=Armand&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCayley1889" class="citation cs2 cs1-prop-long-vol"><a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley, Arthur</a> (1889), <a rel="nofollow" class="external text" href="http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000140">The Collected Mathematical Papers of Arthur Cayley</a>, vol. II (1851–1860), <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Collected+Mathematical+Papers+of+Arthur+Cayley&rft.pub=Cambridge+University+Press&rft.date=1889&rft.aulast=Cayley&rft.aufirst=Arthur&rft_id=http%3A%2F%2Fwww.hti.umich.edu%2Fcgi%2Ft%2Ftext%2Fpageviewer-idx%3Fc%3Dumhistmath%3Bcc%3Dumhistmath%3Brgn%3Dfull%2520text%3Bidno%3DABS3153.0001.001%3Bdidno%3DABS3153.0001.001%3Bview%3Dimage%3Bseq%3D00000140&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFO'ConnorRobertson" class="citation cs2">O'Connor, John J.; <a href="/wiki/Edmund_F._Robertson" class="mw-redirect" title="Edmund F. Robertson">Robertson, Edmund F.</a>, <a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory.html">"The development of group theory"</a>, <a href="/wiki/MacTutor_History_of_Mathematics_Archive" title="MacTutor History of Mathematics Archive">MacTutor History of Mathematics Archive</a>, <a href="/wiki/University_of_St_Andrews" title="University of St Andrews">University of St Andrews</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+development+of+group+theory&rft.btitle=MacTutor+History+of+Mathematics+Archive&rft.pub=University+of+St+Andrews&rft.aulast=O%27Connor&rft.aufirst=John+J.&rft.au=Robertson%2C+Edmund+F.&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FDevelopment_group_theory.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCurtis2003" class="citation cs2"><a href="/wiki/Charles_W._Curtis" title="Charles W. Curtis">Curtis, Charles W.</a> (2003), Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer, History of Mathematics, Providence, R.I.: American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-2677-5" title="Special:BookSources/978-0-8218-2677-5"><bdi>978-0-8218-2677-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pioneers+of+Representation+Theory%3A+Frobenius%2C+Burnside%2C+Schur%2C+and+Brauer&rft.place=Providence%2C+R.I.&rft.series=History+of+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2003&rft.isbn=978-0-8218-2677-5&rft.aulast=Curtis&rft.aufirst=Charles+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFvon_Dyck1882" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Walther_von_Dyck" title="Walther von Dyck">von Dyck, Walther</a> (1882), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140222213905/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0020&DMDID=DMDLOG_0007&L=1">"Gruppentheoretische Studien (Group-theoretical studies)"</a>, <a href="/wiki/Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a> (in German), 20 (1): 1–44, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01443322">10.1007/BF01443322</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:179178038">179178038</a>, archived from <a rel="nofollow" class="external text" href="http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0020&DMDID=DMDLOG_0007&L=1">the original</a> on 2014-02-22</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Gruppentheoretische+Studien+%28Group-theoretical+studies%29&rft.volume=20&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E44&rft.date=1882&rft_id=info%3Adoi%2F10.1007%2FBF01443322&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A179178038%23id-name%3DS2CID&rft.aulast=von+Dyck&rft.aufirst=Walther&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DPPN235181684_0020%26DMDID%3DDMDLOG_0007%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFGalois1908" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Galois, Évariste</a> (1908), Tannery, Jules (ed.), <a rel="nofollow" class="external text" href="http://quod.lib.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=AAN9280">Manuscrits de Évariste Galois [Évariste Galois' Manuscripts]</a> (in French), Paris: Gauthier-Villars</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manuscrits+de+%C3%89variste+Galois+%5B%C3%89variste+Galois%27+Manuscripts%5D&rft.place=Paris&rft.pub=Gauthier-Villars&rft.date=1908&rft.aulast=Galois&rft.aufirst=%C3%89variste&rft_id=http%3A%2F%2Fquod.lib.umich.edu%2Fcgi%2Ft%2Ftext%2Ftext-idx%3Fc%3Dumhistmath%3Bidno%3DAAN9280&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"> (Galois work was first published by <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Joseph Liouville</a> in 1843).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFJordan1870" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan, Camille</a> (1870), <a rel="nofollow" class="external text" href="https://archive.org/details/traitdessubstit00jordgoog">Traité des substitutions et des équations algébriques [Study of Substitutions and Algebraic Equations]</a> (in French), Paris: Gauthier-Villars</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trait%C3%A9+des+substitutions+et+des+%C3%A9quations+alg%C3%A9briques+%5BStudy+of+Substitutions+and+Algebraic+Equations%5D&rft.place=Paris&rft.pub=Gauthier-Villars&rft.date=1870&rft.aulast=Jordan&rft.aufirst=Camille&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftraitdessubstit00jordgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKleiner1986" class="citation cs2"><a href="/wiki/Israel_Kleiner_(mathematician)" title="Israel Kleiner (mathematician)">Kleiner, Israel</a> (1986), "The evolution of group theory: A brief survey", <a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a>, 59 (4): 195–215, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690312">10.2307/2690312</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690312">2690312</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0863090">0863090</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=The+evolution+of+group+theory%3A+A+brief+survey&rft.volume=59&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E195-%3C%2Fspan%3E215&rft.date=1986&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D863090%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2690312%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2690312&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFLie1973" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Sophus_Lie" title="Sophus Lie">Lie, Sophus</a> (1973), Gesammelte Abhandlungen. Band 1 [Collected papers. Volume 1] (in German), New York: Johnson Reprint Corp., <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0392459">0392459</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gesammelte+Abhandlungen.+Band+1+%5BCollected+papers.+Volume+1%5D&rft.place=New+York&rft.pub=Johnson+Reprint+Corp.&rft.date=1973&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0392459%23id-name%3DMR&rft.aulast=Lie&rft.aufirst=Sophus&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMackey1976" class="citation cs2"><a href="/wiki/George_Mackey" title="George Mackey">Mackey, George Whitelaw</a> (1976), The Theory of Unitary Group Representations, <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0396826">0396826</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Unitary+Group+Representations&rft.pub=University+of+Chicago+Press&rft.date=1976&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0396826%23id-name%3DMR&rft.aulast=Mackey&rft.aufirst=George+Whitelaw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith1906" class="citation cs2"><a href="/wiki/David_Eugene_Smith" title="David Eugene Smith">Smith, David Eugene</a> (1906), <a rel="nofollow" class="external text" href="https://www.gutenberg.org/ebooks/8746">History of Modern Mathematics</a>, Mathematical Monographs, No. 1</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=History+of+Modern+Mathematics&rft.series=Mathematical+Monographs%2C+No.+1&rft.date=1906&rft.aulast=Smith&rft.aufirst=David+Eugene&rft_id=https%3A%2F%2Fwww.gutenberg.org%2Febooks%2F8746&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeyl1950" class="citation cs2"><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Weyl, Hermann</a> (1950) [1931], The Theory of Groups and Quantum Mechanics, translated by Robertson, H. P., Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-60269-1" title="Special:BookSources/978-0-486-60269-1"><bdi>978-0-486-60269-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Groups+and+Quantum+Mechanics&rft.pub=Dover&rft.date=1950&rft.isbn=978-0-486-60269-1&rft.aulast=Weyl&rft.aufirst=Hermann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWussing2007" class="citation cs2"><a href="/wiki/Hans_Wussing" title="Hans Wussing">Wussing, Hans</a> (2007), The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory, New York: <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-45868-7" title="Special:BookSources/978-0-486-45868-7"><bdi>978-0-486-45868-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Genesis+of+the+Abstract+Group+Concept%3A+A+Contribution+to+the+History+of+the+Origin+of+Abstract+Group+Theory&rft.place=New+York&rft.pub=Dover+Publications&rft.date=2007&rft.isbn=978-0-486-45868-7&rft.aulast=Wussing&rft.aufirst=Hans&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988">.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Group_(mathematics)&action=edit&section=42" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeisstein" class="citation web cs2"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a>, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Group.html">"Group"</a>, <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Group&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGroup.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+%28mathematics%29" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Groups256" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background: #ceecee"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_navbox" title="Template:Group navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_navbox" title="Template talk:Group navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_navbox" title="Special:EditPage/Template:Group navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Groups256" style="font-size:114%;margin:0 4em"><a class="mw-selflink selflink">Groups</a></div></th></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Group_theory" title="Group theory">Basic notions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Commutator_subgroup" title="Commutator subgroup">Commutator subgroup</a></li> <li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphism</a></li> <li>(<a href="/wiki/Semidirect_product" title="Semidirect product">Semi-</a>) <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">direct sum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%">Types of groups</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">Abelian groups</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic groups</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">Infinite group</a></li> <li><a href="/wiki/Simple_group" title="Simple group">Simple groups</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">Solvable groups</a></li> <li><a href="/wiki/Symmetry_group" title="Symmetry group">Symmetry group</a></li> <li><a href="/wiki/Space_group" title="Space group">Space group</a></li> <li><a href="/wiki/Point_group" title="Point group">Point group</a></li> <li><a href="/wiki/Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li> <li><a href="/wiki/Trivial_group" title="Trivial group">Trivial group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></dt> <dd><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Zn</dd> <dd><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> An</dd></dl> <dl><dt><a href="/wiki/Sporadic_group" title="Sporadic group">Sporadic groups</a></dt> <dd><a href="/wiki/Mathieu_group" title="Mathieu group">Mathieu group</a> M11..12,M22..24</dd> <dd><a href="/wiki/Conway_group" title="Conway group">Conway group</a> Co1..3</dd> <dd>Janko groups <a href="/wiki/Janko_group_J1" title="Janko group J1">J1</a>, <a href="/wiki/Janko_group_J2" title="Janko group J2">J2</a>, <a href="/wiki/Janko_group_J3" title="Janko group J3">J3</a>, <a href="/wiki/Janko_group_J4" title="Janko group J4">J4</a></dd> <dd><a href="/wiki/Fischer_group" title="Fischer group">Fischer group</a> F22..24</dd> <dd><a href="/wiki/Baby_monster_group" title="Baby monster group">Baby monster group</a> B</dd> <dd><a href="/wiki/Monster_group" title="Monster group">Monster group</a> M</dd></dl> <dl><dt>Other finite groups</dt> <dd><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> Sn</dd> <dd><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> Dn</dd> <dd><a href="/wiki/Rubik%27s_Cube_group" title="Rubik's Cube group">Rubik's Cube group</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Lie_group" title="Lie group">Lie groups</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear group</a> GL(n)</li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear group</a> SL(n)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal group</a> O(n)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Special orthogonal group</a> SO(n)</li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary group</a> U(n)</li> <li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary group</a> SU(n)</li> <li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic group</a> Sp(n)</li></ul> <dl><dt><a href="/wiki/Exceptional_Lie_groups" class="mw-redirect" title="Exceptional Lie groups">Exceptional Lie groups</a></dt> <dd><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G2</a></dd> <dd><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F4</a></dd> <dd><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E6</a></dd> <dd><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E7</a></dd> <dd><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E8</a></dd></dl> <ul><li><a href="/wiki/Circle_group" title="Circle group">Circle group</a></li> <li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%">Infinite dimensional groups</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism group</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop group</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li>O(∞)</li> <li>SU(∞)</li> <li>Sp(∞)</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="background: #ceecee;font-weight:bold;"><div> <ul><li><a href="/wiki/History_of_group_theory" title="History of group theory">History</a></li> <li><a href="/wiki/Group_theory#Applications_of_group_theory" title="Group theory">Applications</a></li> <li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /></div><div role="navigation" class="navbox" aria-labelledby="Algebra304" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebra" title="Template:Algebra"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebra" title="Template talk:Algebra"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebra" title="Special:EditPage/Template:Algebra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Algebra304" style="font-size:114%;margin:0 4em"><a href="/wiki/Algebra" title="Algebra">Algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/Outline_of_algebra" title="Outline of algebra">Outline</a></li> <li><a href="/wiki/History_of_algebra" title="History of algebra">History</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Areas</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li> <li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> <ul><li><a href="/wiki/Algebraic_variety" title="Algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">Scheme</a></li></ul></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary algebra</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li> <li><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_expression" title="Algebraic expression">Algebraic expression</a></li> <li><a href="/wiki/Equation" title="Equation">Equation</a> (<a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a>, <a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a>)</li> <li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a> (<a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">Polynomial function</a>)</li> <li><a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">Inequality</a> (<a href="/wiki/Linear_inequality" title="Linear inequality">Linear inequality</a>)</li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> (<a href="/wiki/Addition" title="Addition">Addition</a>, <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a>)</li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a>)</li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">Variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> (<a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">theory</a>)</li> <li><a class="mw-selflink selflink">Group</a> (<a href="/wiki/Group_theory" title="Group theory">theory</a>)</li> <li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a> (<a href="/wiki/Commutative_algebra" title="Commutative algebra">theory</a>)</li> <li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a> (<a href="/wiki/Ring_theory" title="Ring theory">theory</a>)</li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a> (<a href="/wiki/Coordinate_vector" title="Coordinate vector">Vector</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Linear and multilinear algebra</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a> (<a href="/wiki/Dot_product" title="Dot product">Dot product</a>)</li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a> (<a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a>)</li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Linear subspace</a> (<a href="/wiki/Affine_space" title="Affine space">Affine space</a>)</li> <li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">Norm</a> (<a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>)</li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a>)</li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">Trace</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Free_object" title="Free object">Free object</a> (<a href="/wiki/Free_group" title="Free group">Free group</a>, ...)</li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a> (<a href="/wiki/Multivector" title="Multivector">Multivector</a>)</li> <li><a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a> (<a href="/wiki/Polynomial" title="Polynomial">Polynomial</a>)</li> <li><a href="/wiki/Quotient_object" class="mw-redirect" title="Quotient object">Quotient object</a> (<a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a>, ...)</li> <li><a href="/wiki/Symmetric_algebra" title="Symmetric algebra">Symmetric algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topic lists</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category:Algebraic_structures" title="Category:Algebraic structures">Algebraic structures</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Glossaries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Glossary_of_field_theory" title="Glossary of field theory">Field theory</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Linear algebra</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Order theory</a></li> <li><a href="/wiki/Glossary_of_ring_theory" title="Glossary of ring theory">Ring theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Algebra" title="Category:Algebra">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319" /></div><div role="navigation" class="navbox authority-control" aria-label="Navbox673" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <a href="https://www.wikidata.org/wiki/Q83478#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4022379-6">Germany</a></li><li><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph180740&CON_LNG=ENG">Czech Republic</a></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Group_(mathematics)&oldid=1280089198">https://en.wikipedia.org/w/index.php?title=Group_(mathematics)&oldid=1280089198</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Group_theory" title="Category:Group theory">Group theory</a></li><li><a href="/wiki/Category:Algebraic_structures" title="Category:Algebraic structures">Algebraic structures</a></li><li><a href="/wiki/Category:Symmetry" title="Category:Symmetry">Symmetry</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Use_shortened_footnotes_from_September_2024" title="Category:Use shortened footnotes from September 2024">Use shortened footnotes from September 2024</a></li><li><a href="/wiki/Category:Articles_containing_French-language_text" title="Category:Articles containing French-language text">Articles containing French-language text</a></li><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li><li><a href="/wiki/Category:Articles_containing_German-language_text" title="Category:Articles containing German-language text">Articles containing German-language text</a></li><li><a href="/wiki/Category:CS1:_long_volume_value" title="Category:CS1: long volume value">CS1: long volume value</a></li><li><a href="/wiki/Category:Featured_articles" title="Category:Featured articles">Featured articles</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 12 March 2025, at 12:34 (UTC).</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Group_(mathematics)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://www.wikimedia.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" lang="en" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"> Search </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" >Group (mathematics)</div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"> 84 languages </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"> Add topic </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-585dc88b6-967sv","wgBackendResponseTime":433,"wgPageParseReport":{"limitreport":{"cputime":"2.660","walltime":"3.326","ppvisitednodes":{"value":18321,"limit":1000000},"postexpandincludesize":{"value":312204,"limit":2097152},"templateargumentsize":{"value":18653,"limit":2097152},"expansiondepth":{"value":13,"limit":100},"expensivefunctioncount":{"value":26,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":356858,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1836.765 1 -total"," 24.70% 453.622 75 Template:Citation"," 17.29% 317.590 90 Template:Sfn"," 7.58% 139.249 2 Template:Sidebar_with_collapsible_lists"," 7.25% 133.112 3 Template:Lang"," 7.16% 131.455 1 Template:Group_theory_sidebar"," 6.80% 124.985 1 Template:Short_description"," 4.86% 89.231 21 Template:Efn"," 4.74% 87.072 118 Template:Main_other"," 4.53% 83.157 2 Template:Pagetype"]},"scribunto":{"limitreport-timeusage":{"value":"1.102","limit":"10.000"},"limitreport-memusage":{"value":17960571,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREF\"] = 3,\n [\"CITEREFArtin1998\"] = 1,\n [\"CITEREFArtin2018\"] = 1,\n [\"CITEREFAschbacher2004\"] = 1,\n [\"CITEREFAwodey2010\"] = 1,\n [\"CITEREFBehlerWicklederChristoffers2014\"] = 1,\n [\"CITEREFBersuker2006\"] = 1,\n [\"CITEREFBescheEickO\u0026#039;Brien2001\"] = 1,\n [\"CITEREFBishop1993\"] = 1,\n [\"CITEREFBorel1991\"] = 1,\n [\"CITEREFBorel2001\"] = 1,\n [\"CITEREFCarter1989\"] = 1,\n [\"CITEREFCayley1889\"] = 1,\n [\"CITEREFChanceyO\u0026#039;Brien2021\"] = 1,\n [\"CITEREFConwayDelgado_FriedrichsHusonThurston2001\"] = 1,\n [\"CITEREFCook2009\"] = 1,\n [\"CITEREFCoornaertDelzantPapadopoulos1990\"] = 1,\n [\"CITEREFCurtis2003\"] = 1,\n [\"CITEREFDeneckeWismath2002\"] = 1,\n [\"CITEREFDove2003\"] = 1,\n [\"CITEREFDudek2001\"] = 1,\n [\"CITEREFElielWilenMander1994\"] = 1,\n [\"CITEREFEllis2019\"] = 1,\n [\"CITEREFFrucht1939\"] = 1,\n [\"CITEREFFultonHarris1991\"] = 1,\n [\"CITEREFGalois1908\"] = 1,\n [\"CITEREFGoldstein1980\"] = 1,\n [\"CITEREFGollmann2011\"] = 1,\n [\"CITEREFHall1967\"] = 1,\n [\"CITEREFHatcher2002\"] = 1,\n [\"CITEREFHerstein1975\"] = 1,\n [\"CITEREFHerstein1996\"] = 1,\n [\"CITEREFHusain1966\"] = 1,\n [\"CITEREFJahnTeller1937\"] = 1,\n [\"CITEREFJordan1870\"] = 1,\n [\"CITEREFKleiner1986\"] = 1,\n [\"CITEREFKuga1993\"] = 1,\n [\"CITEREFKuipers1999\"] = 1,\n [\"CITEREFKurzweilStellmacher2004\"] = 1,\n [\"CITEREFLang2005\"] = 1,\n [\"CITEREFLay2003\"] = 1,\n [\"CITEREFLedermann1953\"] = 1,\n [\"CITEREFLedermann1973\"] = 1,\n [\"CITEREFLie1973\"] = 1,\n [\"CITEREFMac_Lane1998\"] = 1,\n [\"CITEREFMackey1976\"] = 1,\n [\"CITEREFMagnusKarrassSolitar2004\"] = 1,\n [\"CITEREFMathSciNet2021\"] = 1,\n [\"CITEREFMichler2006\"] = 1,\n [\"CITEREFMilne1980\"] = 1,\n [\"CITEREFMumfordFogartyKirwan1994\"] = 1,\n [\"CITEREFNaber2003\"] = 1,\n [\"CITEREFNeukirch1999\"] = 1,\n [\"CITEREFRobinson1996\"] = 1,\n [\"CITEREFRomanowskaSmith2002\"] = 1,\n [\"CITEREFRonan2007\"] = 1,\n [\"CITEREFRosen2000\"] = 1,\n [\"CITEREFRudin1990\"] = 1,\n [\"CITEREFSchwartzman1994\"] = 1,\n [\"CITEREFSeress1997\"] = 1,\n [\"CITEREFSerre1977\"] = 1,\n [\"CITEREFShatz1972\"] = 1,\n [\"CITEREFSimons2003\"] = 1,\n [\"CITEREFSmith1906\"] = 1,\n [\"CITEREFSolomon2018\"] = 1,\n [\"CITEREFStewart2015\"] = 1,\n [\"CITEREFSuzuki1951\"] = 1,\n [\"CITEREFWarner1983\"] = 1,\n [\"CITEREFWeinberg1972\"] = 1,\n [\"CITEREFWelsh1989\"] = 1,\n [\"CITEREFWeyl1950\"] = 1,\n [\"CITEREFWeyl1952\"] = 1,\n [\"CITEREFWussing2007\"] = 1,\n [\"CITEREFZee2010\"] = 1,\n [\"CITEREFvon_Dyck1882\"] = 1,\n}\ntemplate_list = table#1 {\n [\"About\"] = 1,\n [\"Algebra\"] = 1,\n [\"Algebraic structures\"] = 1,\n [\"Authority control\"] = 1,\n [\"Block indent\"] = 2,\n [\"Br\"] = 19,\n [\"Citation\"] = 74,\n [\"Clear\"] = 3,\n [\"Color box\"] = 4,\n [\"DEFAULTSORT:Group (Mathematics)\"] = 1,\n [\"Efn\"] = 21,\n [\"Featured article\"] = 1,\n [\"Group navbox\"] = 1,\n [\"Group theory sidebar\"] = 1,\n [\"Group-like structures\"] = 1,\n [\"Harvard citations\"] = 1,\n [\"Harvnb\"] = 22,\n [\"Hatnote\"] = 1,\n [\"Lang\"] = 3,\n [\"Lang Algebra\"] = 1,\n [\"MacTutor\"] = 1,\n [\"Main\"] = 15,\n [\"Math\"] = 18,\n [\"MathWorld\"] = 1,\n [\"Mvar\"] = 5,\n [\"No\"] = 13,\n [\"Notelist\"] = 1,\n [\"Nowrap\"] = 1,\n [\"Portal\"] = 1,\n [\"Quote box\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n [\"See also\"] = 2,\n [\"Sfn\"] = 90,\n [\"Sfrac\"] = 1,\n [\"Short description\"] = 1,\n [\"TOClimit\"] = 1,\n [\"Tmath\"] = 233,\n [\"Use shortened footnotes\"] = 1,\n [\"Weibel IHA\"] = 1,\n [\"Yes\"] = 25,\n}\narticle_whitelist = table#1 {\n}\nciteref_patterns = table#1 {\n}\n","limitreport-profile":[["?","260","23.2"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::callParserFunction","200","17.9"],["dataWrapper \u003Cmw.lua:672\u003E","120","10.7"],["type","80","7.1"],["recursiveClone \u003CmwInit.lua:45\u003E","60","5.4"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::getAllExpandedArguments","60","5.4"],["concat","40","3.6"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::gsub","40","3.6"],["MediaWiki\\Extension\\Scribunto\\Engines\\LuaSandbox\\LuaSandboxCallback::sub","40","3.6"],["select_one \u003CModule:Citation/CS1/Utilities:429\u003E","20","1.8"],["[others]","200","17.9"]]},"cachereport":{"origin":"mw-web.eqiad.main-65bf7dbd64-qzsht","timestamp":"20250325193031","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Group (mathematics)","url":"https:\/\/en.wikipedia.org\/wiki\/Group_(mathematics)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q83478","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q83478","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2001-11-16T13:58:05Z","dateModified":"2025-03-12T12:34:54Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/a\/a6\/Rubik%27s_cube.svg","headline":"algebraic set with an invertible, associative internal operation admitting a neutral element"}</script> </body> </html>