CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Group action - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-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":"c23a3869-994d-468d-80eb-d9b316a67f9c","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Group_action","wgTitle":"Group action","wgCurRevisionId":1256994680,"wgRevisionId":1256994680,"wgArticleId":12781,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","All articles with unsourced statements","Articles with unsourced statements from May 2023","All accuracy disputes","Articles with disputed statements from March 2015","Group theory","Group actions (mathematics)","Representation theory of groups","Symmetry"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Group_action","wgRelevantArticleId":12781, "wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Orbit_(group_theory)","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":50000,"wgInternalRedirectTargetUrl":"/wiki/Group_action#Orbits_and_stabilizers","wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q288465", "wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":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","mediawiki.page.gallery.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","wikibase.sidebar.tracking"];</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%7Cmediawiki.page.gallery.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.4"> <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/5/5f/Cyclic_group.svg/1200px-Cyclic_group.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1167"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/800px-Cyclic_group.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="778"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/640px-Cyclic_group.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="623"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Group action - 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_action#Orbits_and_stabilizers"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Group_action&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_action#Orbits_and_stabilizers"> <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_action rootpage-Group_action 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" > <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> </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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_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+action" 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+action" 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/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_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+action" 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+action" 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" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 1 Definition </div> </a> <button aria-controls="toc-Definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Definition subsection </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Left_group_action" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Left_group_action"> <div class="vector-toc-text"> 1.1 Left group action </div> </a> <ul id="toc-Left_group_action-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Right_group_action" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Right_group_action"> <div class="vector-toc-text"> 1.2 Right group action </div> </a> <ul id="toc-Right_group_action-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notable_properties_of_actions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notable_properties_of_actions"> <div class="vector-toc-text"> 2 Notable properties of actions </div> </a> <button aria-controls="toc-Notable_properties_of_actions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Notable properties of actions subsection </button> <ul id="toc-Notable_properties_of_actions-sublist" class="vector-toc-list"> <li id="toc-Transitivity_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transitivity_properties"> <div class="vector-toc-text"> 2.1 Transitivity properties </div> </a> <ul id="toc-Transitivity_properties-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 2.1.1 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Primitive_actions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primitive_actions"> <div class="vector-toc-text"> 2.2 Primitive actions </div> </a> <ul id="toc-Primitive_actions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_properties"> <div class="vector-toc-text"> 2.3 Topological properties </div> </a> <ul id="toc-Topological_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Actions_of_topological_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Actions_of_topological_groups"> <div class="vector-toc-text"> 2.4 Actions of topological groups </div> </a> <ul id="toc-Actions_of_topological_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Linear_actions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_actions"> <div class="vector-toc-text"> 2.5 Linear actions </div> </a> <ul id="toc-Linear_actions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Orbits_and_stabilizers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Orbits_and_stabilizers"> <div class="vector-toc-text"> 3 Orbits and stabilizers </div> </a> <button aria-controls="toc-Orbits_and_stabilizers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Orbits and stabilizers subsection </button> <ul id="toc-Orbits_and_stabilizers-sublist" class="vector-toc-list"> <li id="toc-Invariant_subsets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Invariant_subsets"> <div class="vector-toc-text"> 3.1 Invariant subsets </div> </a> <ul id="toc-Invariant_subsets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fixed_points_and_stabilizer_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fixed_points_and_stabilizer_subgroups"> <div class="vector-toc-text"> 3.2 Fixed points and stabilizer subgroups </div> </a> <ul id="toc-Fixed_points_and_stabilizer_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Orbit-stabilizer_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Orbit-stabilizer_theorem"> <div class="vector-toc-text"> 3.3 Orbit-stabilizer theorem </div> </a> <ul id="toc-Orbit-stabilizer_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Burnside's_lemma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Burnside's_lemma"> <div class="vector-toc-text"> 3.4 Burnside's lemma </div> </a> <ul id="toc-Burnside's_lemma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples_2" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_2"> <div class="vector-toc-text"> 4 Examples </div> </a> <ul id="toc-Examples_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_actions_and_groupoids" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Group_actions_and_groupoids"> <div class="vector-toc-text"> 5 Group actions and groupoids </div> </a> <ul id="toc-Group_actions_and_groupoids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Morphisms_and_isomorphisms_between_G-sets" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Morphisms_and_isomorphisms_between_G-sets"> <div class="vector-toc-text"> 6 Morphisms and isomorphisms between G-sets </div> </a> <ul id="toc-Morphisms_and_isomorphisms_between_G-sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variants_and_generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Variants_and_generalizations"> <div class="vector-toc-text"> 7 Variants and generalizations </div> </a> <ul id="toc-Variants_and_generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Gallery" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Gallery"> <div class="vector-toc-text"> 8 Gallery </div> </a> <ul id="toc-Gallery-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <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 vector-toc-list-item-expanded"> <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 vector-toc-list-item-expanded"> <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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <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" > <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 action</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 25 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-25" aria-hidden="true" > 25 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B5%D0%B9%D1%81%D1%82%D0%B2%D0%B8%D0%B5_%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Действие на група – Bulgarian" lang="bg" hreflang="bg" data-title="Действие на група" 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/Acci%C3%B3_(matem%C3%A0tiques)" title="Acció (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Acció (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Akce_grupy_na_mno%C5%BEin%C4%9B" title="Akce grupy na množině – Czech" lang="cs" hreflang="cs" data-title="Akce grupy na množině" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gruppenoperation" title="Gruppenoperation – German" lang="de" hreflang="de" data-title="Gruppenoperation" 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%BChma_toime" title="Rühma toime – Estonian" lang="et" hreflang="et" data-title="Rühma toime" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Acci%C3%B3n_(matem%C3%A1tica)" title="Acción (matemática) – Spanish" lang="es" hreflang="es" data-title="Acción (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/Grupa_ago" title="Grupa ago – Esperanto" lang="eo" hreflang="eo" data-title="Grupa ago" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%A9%D9%86%D8%B4_%DA%AF%D8%B1%D9%88%D9%87%DB%8C" title="کنش گروهی – Persian" lang="fa" hreflang="fa" data-title="کنش گروهی" 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/Action_de_groupe_(math%C3%A9matiques)" title="Action de groupe (mathématiques) – French" lang="fr" hreflang="fr" data-title="Action de groupe (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0%EC%9D%98_%EC%9E%91%EC%9A%A9" title="군의 작용 – Korean" lang="ko" hreflang="ko" data-title="군의 작용" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Tindakan_grup_(matematika)" title="Tindakan grup (matematika) – Indonesian" lang="id" hreflang="id" data-title="Tindakan 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Azione_di_gruppo" title="Azione di gruppo – Italian" lang="it" hreflang="it" data-title="Azione di gruppo" 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%A4%D7%A2%D7%95%D7%9C%D7%AA_%D7%97%D7%91%D7%95%D7%A8%D7%94" title="פעולת חבורה – Hebrew" lang="he" hreflang="he" data-title="פעולת חבורה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Csoporthat%C3%A1s" title="Csoporthatás – Hungarian" lang="hu" hreflang="hu" data-title="Csoporthatás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Groepswerking" title="Groepswerking – Dutch" lang="nl" hreflang="nl" data-title="Groepswerking" 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%E4%BD%9C%E7%94%A8" title="群作用 – Japanese" lang="ja" hreflang="ja" data-title="群作用" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Assion" title="Assion – Piedmontese" lang="pms" hreflang="pms" data-title="Assion" 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/Dzia%C5%82anie_grupy_na_zbiorze" title="Działanie grupy na zbiorze – Polish" lang="pl" hreflang="pl" data-title="Działanie grupy na zbiorze" 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/A%C3%A7%C3%A3o_de_grupo" title="Ação de grupo – Portuguese" lang="pt" hreflang="pt" data-title="Ação de grupo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B5%D0%B9%D1%81%D1%82%D0%B2%D0%B8%D0%B5_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D1%8B" title="Действие группы – Russian" lang="ru" hreflang="ru" data-title="Действие группы" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Gruppverkan" title="Gruppverkan – Swedish" lang="sv" hreflang="sv" data-title="Gruppverkan" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D1%96%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%B8" title="Дія групи – Ukrainian" lang="uk" hreflang="uk" data-title="Дія групи" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%A1c_%C4%91%E1%BB%99ng_nh%C3%B3m" title="Tác động nhóm – Vietnamese" lang="vi" hreflang="vi" data-title="Tác động nhóm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%BE%A3%E4%BD%9C%E7%94%A8" title="羣作用 – Cantonese" lang="yue" hreflang="yue" data-title="羣作用" 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%E4%BD%9C%E7%94%A8" title="群作用 – Chinese" lang="zh" hreflang="zh" data-title="群作用" 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/Q288465#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_action" 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_action" 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_action">Read</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Group_action&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_action&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_action">Read</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Group_action&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_action&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_action" 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_action" 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="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u">Upload file</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q">Special pages</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Group_action&oldid=1256994680" 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_action&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_action&id=1256994680&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_action%23Orbits_and_stabilizers">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_action%23Orbits_and_stabilizers">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_action&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_action&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_actions" 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_actions_on_sets" 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/Q288465" 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> <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=Orbit_(group_theory)&redirect=no" class="mw-redirect" title="Orbit (group theory)">Orbit (group theory)</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">Transformations induced by a mathematical group</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 the mathematical concept. For the sociology term, see <a href="/wiki/Group_action_(sociology)" title="Group action (sociology)">group action (sociology)</a>.</div> <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 ); 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 class="mw-selflink selflink">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 ); 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> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Group_action_on_equilateral_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Group_action_on_equilateral_triangle.svg/220px-Group_action_on_equilateral_triangle.svg.png" decoding="async" width="220" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Group_action_on_equilateral_triangle.svg/330px-Group_action_on_equilateral_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Group_action_on_equilateral_triangle.svg/440px-Group_action_on_equilateral_triangle.svg.png 2x" data-file-width="467" data-file-height="485" /></a><figcaption>The <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> C3 consisting of the <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> by 0°, 120° and 240° acts on the set of the three vertices.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, many sets of <a href="/wiki/Transformation_(function)" title="Transformation (function)">transformations</a> form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under <a href="/wiki/Function_composition" title="Function composition">function composition</a>; for example, the <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotations</a> around a point in the plane. It is often useful to consider the group as an <a href="/wiki/Abstract_group" class="mw-redirect" title="Abstract group">abstract group</a>, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a <a href="/wiki/Mathematical_structure" title="Mathematical structure">structure</a> acts also on various related structures; for example, the above rotation group acts also on triangles by transforming triangles into triangles. Formally, a group action of a group G on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> S is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> from G to some group (under <a href="/wiki/Function_composition" title="Function composition">function composition</a>) of functions from S to itself. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of <a href="/wiki/Euclidean_isometry" class="mw-redirect" title="Euclidean isometry">Euclidean isometries</a> acts on <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> and also on the figures drawn in it; in particular, it acts on the set of all <a href="/wiki/Triangle" title="Triangle">triangles</a>. Similarly, the group of <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a> of a <a href="/wiki/Polyhedron" title="Polyhedron">polyhedron</a> acts on the <a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">vertices</a>, the <a href="/wiki/Edge_(geometry)" title="Edge (geometry)">edges</a>, and the <a href="/wiki/Face_(geometry)" title="Face (geometry)">faces</a> of the polyhedron. A group action on a <a href="/wiki/Vector_space" title="Vector space">vector space</a> is called a <a href="/wiki/Group_representation" title="Group representation">representation</a> of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with <a href="/wiki/Subgroups" class="mw-redirect" title="Subgroups">subgroups</a> of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> GL(n, K), the group of the <a href="/wiki/Invertible_matrices" class="mw-redirect" title="Invertible matrices">invertible matrices</a> of <a href="/wiki/Dimension" title="Dimension">dimension</a> n over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> K. The <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> Sn acts on any <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> with n elements by permuting the elements of the set. Although the group of all <a href="/wiki/Permutation" title="Permutation">permutations</a> of a set depends formally on the set, the concept of group action allows one to consider a single group for studying the permutations of all sets with the same <a href="/wiki/Cardinality" title="Cardinality">cardinality</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Left_group_action">Left group action</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=2" title="Edit section: Left group action">edit</a>]</div> If G is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> with <a href="/wiki/Identity_element" title="Identity element">identity element</a> e, and X is a set, then a (left) group action α of G on X is a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon G\times X\to X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>G</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \colon G\times X\to X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf7b5dab5b12bccc26aba8d70cd10f033235207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.41ex; height:2.509ex;" alt="{\displaystyle \alpha \colon G\times X\to X,}"></dd></dl> that satisfies the following two <a href="/wiki/Axioms" class="mw-redirect" title="Axioms">axioms</a>:<a href="#cite_note-1">[1]</a> <dl><dd><table> <tbody><tr> <td>Identity: </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (e,x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>e</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (e,x)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3561976c1831704a750238f40e9f8226c6445282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.172ex; height:2.843ex;" alt="{\displaystyle \alpha (e,x)=x}"> </td></tr> <tr> <td>Compatibility: </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (g,\alpha (h,x))=\alpha (gh,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mi>h</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (g,\alpha (h,x))=\alpha (gh,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03f8d33310c99c29d66c7443ac84617a26a21daf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.66ex; height:2.843ex;" alt="{\displaystyle \alpha (g,\alpha (h,x))=\alpha (gh,x)}"> </td></tr></tbody></table></dd></dl> for all g and h in G and all x in X. The group G is then said to act on X (from the left). A set X together with an action of G is called a (left) G-set. It can be notationally convenient to <a href="/wiki/Currying" title="Currying">curry</a> the action α, so that, instead, one has a collection of <a href="/wiki/Transformation_(geometry)" class="mw-redirect" title="Transformation (geometry)">transformations</a> αg : X → X, with one transformation αg for each group element g ∈ G. The identity and compatibility relations then read <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{e}(x)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{e}(x)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d879f406b55745b0266e93fa0993c36dacbeb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.053ex; height:2.843ex;" alt="{\displaystyle \alpha _{e}(x)=x}"></dd></dl> and <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> <mo>∘</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mi>h</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cc509b62b21ace4715c3a05c229cfacca89065d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.234ex; height:3.009ex;" alt="{\displaystyle \alpha _{g}(\alpha _{h}(x))=(\alpha _{g}\circ \alpha _{h})(x)=\alpha _{gh}(x)}"></dd></dl> with ∘ being <a href="/wiki/Function_composition" title="Function composition">function composition</a>. The second axiom then states that the function composition is compatible with the group multiplication; they form a <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutative diagram</a>. This axiom can be shortened even further, and written as αg ∘ αh = αgh. With the above understanding, it is very common to avoid writing α entirely, and to replace it with either a dot, or with nothing at all. Thus, α(g, x) can be shortened to g⋅x or gx, especially when the action is clear from context. The axioms are then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e{\cdot }x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e{\cdot }x=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9ceb0477ff18e21bd41f3f6c331c03cde4a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.488ex; height:1.676ex;" alt="{\displaystyle e{\cdot }x=x}"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g{\cdot }(h{\cdot }x)=(gh){\cdot }x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mi>h</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g{\cdot }(h{\cdot }x)=(gh){\cdot }x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901503f3f767332e832b09c0b695a49d278c94d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.227ex; height:2.843ex;" alt="{\displaystyle g{\cdot }(h{\cdot }x)=(gh){\cdot }x}"></dd></dl> From these two axioms, it follows that for any fixed g in G, the function from X to itself which maps x to g⋅x is a <a href="/wiki/Bijection" title="Bijection">bijection</a>, with inverse bijection the corresponding map for g−1. Therefore, one may equivalently define a group action of G on X as a group homomorphism from G into the symmetric group Sym(X) of all bijections from X to itself.<a href="#cite_note-2">[2]</a> <div class="mw-heading mw-heading3"><h3 id="Right_group_action">Right group action</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=3" title="Edit section: Right group action">edit</a>]</div> Likewise, a right group action of G on X is a function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon X\times G\to X,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>X</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \colon X\times G\to X,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701b5addd623fafb50926517c30c83813ea9833c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.41ex; height:2.509ex;" alt="{\displaystyle \alpha \colon X\times G\to X,}"></dd></dl> that satisfies the analogous axioms:<a href="#cite_note-3">[3]</a> <dl><dd><table> <tbody><tr> <td>Identity: </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (x,e)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (x,e)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/614802108ff47918f4996e03f7e858131418af8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.172ex; height:2.843ex;" alt="{\displaystyle \alpha (x,e)=x}"> </td></tr> <tr> <td>Compatibility: </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha (\alpha (x,g),h)=\alpha (x,gh)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>g</mi> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha (\alpha (x,g),h)=\alpha (x,gh)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f54205aee0089ec041830d9833001f5a9e0f8fa5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.66ex; height:2.843ex;" alt="{\displaystyle \alpha (\alpha (x,g),h)=\alpha (x,gh)}"> </td></tr></tbody></table></dd></dl> (with α(x, g) often shortened to xg or x⋅g when the action being considered is clear from context) <dl><dd><table> <tbody><tr> <td>Identity: </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x{\cdot }e=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>e</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x{\cdot }e=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ebd8961c2a659a8b3ad1554e444fd5a38cee843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.488ex; height:1.676ex;" alt="{\displaystyle x{\cdot }e=x}"> </td></tr> <tr> <td>Compatibility: </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x{\cdot }g){\cdot }h=x{\cdot }(gh)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>g</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>h</mi> <mo>=</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mi>h</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x{\cdot }g){\cdot }h=x{\cdot }(gh)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48867204316ef06523e85759ae0498a15535130c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.227ex; height:2.843ex;" alt="{\displaystyle (x{\cdot }g){\cdot }h=x{\cdot }(gh)}"> </td></tr></tbody></table></dd></dl> for all g and h in G and all x in X. The difference between left and right actions is in the order in which a product gh acts on x. For a left action, h acts first, followed by g second. For a right action, g acts first, followed by h second. Because of the formula (gh)−1 = h−1g−1, a left action can be constructed from a right action by composing with the inverse operation of the group. Also, a right action of a group G on X can be considered as a left action of its <a href="/wiki/Opposite_group" title="Opposite group">opposite group</a> Gop on X. Thus, for establishing general properties of group actions, it suffices to consider only left actions. However, there are cases where this is not possible. For example, the multiplication of a group <a href="/wiki/Induced_representation" title="Induced representation">induces</a> both a left action and a right action on the group itself—multiplication on the left and on the right, respectively. <div class="mw-heading mw-heading2"><h2 id="Notable_properties_of_actions">Notable properties of actions</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=4" title="Edit section: Notable properties of actions">edit</a>]</div> Let G be a group acting on a set X. The action is called <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style>faithful or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">effective if g⋅x = x for all x ∈ X implies that g = eG. Equivalently, the <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> from G to the group of bijections of X corresponding to the action is <a href="/wiki/Injective" class="mw-redirect" title="Injective">injective</a>. The action is called <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">free (or semiregular or fixed-point free) if the statement that g⋅x = x for some x ∈ X already implies that g = eG. In other words, no non-trivial element of G fixes a point of X. This is a much stronger property than faithfulness. For example, the action of any group on itself by left multiplication is free. This observation implies <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a> that any group can be <a href="/wiki/Embedding" title="Embedding">embedded</a> in a symmetric group (which is infinite when the group is). A finite group may act faithfully on a set of size much smaller than its cardinality (however such an action cannot be free). For instance the abelian 2-group (Z / 2Z)n (of cardinality 2n) acts faithfully on a set of size 2n. This is not always the case, for example the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> Z / 2nZ cannot act faithfully on a set of size less than 2n. In general the smallest set on which a faithful action can be defined can vary greatly for groups of the same size. For example, three groups of size 120 are the symmetric group S5, the icosahedral group A5 × Z / 2Z and the cyclic group Z / 120Z. The smallest sets on which faithful actions can be defined for these groups are of size 5, 7, and 16 respectively. <div class="mw-heading mw-heading3"><h3 id="Transitivity_properties">Transitivity properties</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=5" title="Edit section: Transitivity properties">edit</a>]</div> The action of G on X is called <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">transitive if for any two points x, y ∈ X there exists a g ∈ G so that g ⋅ x = y. The action is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">simply transitive (or sharply transitive, or <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">regular) if it is both transitive and free. This means that given x, y ∈ X the element g in the definition of transitivity is unique. If X is acted upon simply transitively by a group G then it is called a <a href="/wiki/Principal_homogeneous_space" title="Principal homogeneous space">principal homogeneous space</a> for G or a G-torsor. For an integer n ≥ 1, the action is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">n-transitive if X has at least n elements, and for any pair of n-tuples (x1, ..., xn), (y1, ..., yn) ∈ Xn with pairwise distinct entries (that is xi ≠ xj, yi ≠ yj when i ≠ j) there exists a g ∈ G such that g⋅xi = yi for i = 1, ..., n. In other words, the action on the subset of Xn of tuples without repeated entries is transitive. For n = 2, 3 this is often called double, respectively triple, transitivity. The class of <a href="/wiki/2-transitive_group" class="mw-redirect" title="2-transitive group">2-transitive groups</a> (that is, subgroups of a finite symmetric group whose action is 2-transitive) and more generally <a href="/wiki/Multiply_transitive_group" class="mw-redirect" title="Multiply transitive group">multiply transitive groups</a> is well-studied in finite group theory. An action is <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">sharply n-transitive when the action on tuples without repeated entries in Xn is sharply transitive. <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4>[<a href="/w/index.php?title=Group_action&action=edit&section=6" title="Edit section: Examples">edit</a>]</div> The action of the symmetric group of X is transitive, in fact n-transitive for any n up to the cardinality of X. If X has cardinality n, the action of the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> is (n − 2)-transitive but not (n − 1)-transitive. The action of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> of a vector space V on the set V ∖ {0} of non-zero vectors is transitive, but not 2-transitive (similarly for the action of the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> if the dimension of v is at least 2). The action of the <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> of a Euclidean space is not transitive on nonzero vectors but it is on the <a href="/wiki/Unit_sphere" title="Unit sphere">unit sphere</a>. <div class="mw-heading mw-heading3"><h3 id="Primitive_actions">Primitive actions</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=7" title="Edit section: Primitive actions">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/Primitive_permutation_group" title="Primitive permutation group">primitive permutation group</a></div> The action of G on X is called primitive if there is no <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of X preserved by all elements of G apart from the trivial partitions (the partition in a single piece and its <a href="/wiki/Dual_space" title="Dual space">dual</a>, the partition into <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singletons</a>). <div class="mw-heading mw-heading3"><h3 id="Topological_properties">Topological properties</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=8" title="Edit section: Topological properties">edit</a>]</div> Assume that X is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> and the action of G is by <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a>. The action is wandering if every x ∈ X has a <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> U such that there are only finitely many g ∈ G with g⋅U ∩ U ≠ ∅.<a href="#cite_note-FOOTNOTEThurston1997Definition_3.5.1(iv)-4">[4]</a> More generally, a point x ∈ X is called a point of discontinuity for the action of G if there is an open subset U ∋ x such that there are only finitely many g ∈ G with g⋅U ∩ U ≠ ∅. The domain of discontinuity of the action is the set of all points of discontinuity. Equivalently it is the largest G-stable open subset Ω ⊂ X such that the action of G on Ω is wandering.<a href="#cite_note-FOOTNOTEKapovich2009p._73-5">[5]</a> In a dynamical context this is also called a <a href="/wiki/Wandering_set" title="Wandering set">wandering set</a>. The action is properly discontinuous if for every <a href="/wiki/Compact_space" title="Compact space">compact</a> subset K ⊂ X there are only finitely many g ∈ G such that g⋅K ∩ K ≠ ∅. This is strictly stronger than wandering; for instance the action of Z on R2 ∖ {(0, 0)} given by n⋅(x, y) = (2nx, 2−ny) is wandering and free but not properly discontinuous.<a href="#cite_note-FOOTNOTEThurston1980176-6">[6]</a> The action by <a href="/wiki/Deck_transformation" class="mw-redirect" title="Deck transformation">deck transformations</a> of the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> of a locally simply connected space on a <a href="/wiki/Covering_space#Universal_covering" title="Covering space">universal cover</a> is wandering and free. Such actions can be characterized by the following property: every x ∈ X has a neighbourhood U such that g⋅U ∩ U = ∅ for every g ∈ G ∖ {eG}.<a href="#cite_note-FOOTNOTEHatcher2002p._72-7">[7]</a> Actions with this property are sometimes called freely discontinuous, and the largest subset on which the action is freely discontinuous is then called the free regular set.<a href="#cite_note-FOOTNOTEMaskit1988II.A.1,_II.A.2-8">[8]</a> An action of a group G on a <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact space</a> X is called <a href="/wiki/Cocompact_group_action" title="Cocompact group action">cocompact</a> if there exists a compact subset A ⊂ X such that X = G ⋅ A. For a properly discontinuous action, cocompactness is equivalent to compactness of the <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> G \ X. <div class="mw-heading mw-heading3"><h3 id="Actions_of_topological_groups">Actions of topological groups</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=9" title="Edit section: Actions of topological 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/Continuous_group_action" title="Continuous group action">Continuous group action</a></div> Now assume G is a <a href="/wiki/Topological_group" title="Topological group">topological group</a> and X a topological space on which it acts by homeomorphisms. The action is said to be continuous if the map G × X → X is continuous for the <a href="/wiki/Product_topology" title="Product topology">product topology</a>. The action is said to be <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">proper if the map G × X → X × X defined by (g, x) ↦ (x, g⋅x) is <a href="/wiki/Proper_map" title="Proper map">proper</a>.<a href="#cite_note-FOOTNOTEtom_Dieck1987-9">[9]</a> This means that given compact sets K, K′ the set of g ∈ G such that g⋅K ∩ K′ ≠ ∅ is compact. In particular, this is equivalent to proper discontinuity G is a <a href="/wiki/Discrete_group" title="Discrete group">discrete group</a>. It is said to be locally free if there exists a neighbourhood U of eG such that g⋅x ≠ x for all x ∈ X and g ∈ U ∖ {eG}. The action is said to be strongly continuous if the orbital map g ↦ g⋅x is continuous for every x ∈ X. Contrary to what the name suggests, this is a weaker property than continuity of the action.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] If G is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a> and X a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, then the subspace of smooth points for the action is the set of points x ∈ X such that the map g ↦ g⋅x is <a href="/wiki/Smooth_map" class="mw-redirect" title="Smooth map">smooth</a>. There is a well-developed theory of <a href="/wiki/Lie_group_action" title="Lie group action">Lie group actions</a>, i.e. action which are smooth on the whole space. <div class="mw-heading mw-heading3"><h3 id="Linear_actions">Linear actions</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=10" title="Edit section: Linear actions">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_representation" title="Group representation">Group representation</a></div> If g acts by linear transformations on a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a> over a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, the action is said to be <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible</a> if there are no proper nonzero g-invariant submodules. It is said to be <a href="/wiki/Semi-simplicity" title="Semi-simplicity">semisimple</a> if it decomposes as a <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a> of irreducible actions. <div class="mw-heading mw-heading2"><h2 id="Orbits_and_stabilizers"> Orbits and stabilizers</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=11" title="Edit section: Orbits and stabilizers">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Compound_of_five_tetrahedra.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Compound_of_five_tetrahedra.png/220px-Compound_of_five_tetrahedra.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Compound_of_five_tetrahedra.png/330px-Compound_of_five_tetrahedra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6c/Compound_of_five_tetrahedra.png/440px-Compound_of_five_tetrahedra.png 2x" data-file-width="1000" data-file-height="1000" /></a><figcaption>In the <a href="/wiki/Compound_of_five_tetrahedra" title="Compound of five tetrahedra">compound of five tetrahedra</a>, the symmetry group is the (rotational) icosahedral group I of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) <a href="/wiki/Tetrahedral_group" class="mw-redirect" title="Tetrahedral group">tetrahedral group</a> T of order 12, and the orbit space I / T (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset gT corresponds to the tetrahedron to which g sends the chosen tetrahedron.</figcaption></figure> Consider a group G acting on a set X. The <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">orbit of an element x in X is the set of elements in X to which x can be moved by the elements of G. The orbit of x is denoted by G⋅x: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G{\cdot }x=\{g{\cdot }x: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>x</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mo>:</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf07be5473d2e0459718618f2c1f0f380ec880c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.687ex; height:2.843ex;" alt="{\displaystyle G{\cdot }x=\{g{\cdot }x:g\in G\}.}"> The defining properties of a group guarantee that the set of orbits of (points x in) X under the action of G form a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of X. The associated <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> is defined by saying x ~ y <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> there exists a g in G with g⋅x = y. The orbits are then the <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> under this relation; two elements x and y are equivalent if and only if their orbits are the same, that is, G⋅x = G⋅y. The group action is <a href="/wiki/Group_action_(mathematics)#Notable_properties_of_actions" class="mw-redirect" title="Group action (mathematics)">transitive</a> if and only if it has exactly one orbit, that is, if there exists x in X with G⋅x = X. This is the case if and only if G⋅x = X for all x in X (given that X is non-empty). The set of all orbits of X under the action of G is written as X / G (or, less frequently, as G \ X), and is called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">quotient of the action. In geometric situations it may be called the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">orbit space, while in algebraic situations it may be called the space of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">coinvariants, and written XG, by contrast with the invariants (fixed points), denoted XG: the coinvariants are a quotient while the invariants are a subset. The coinvariant terminology and notation are used particularly in <a href="/wiki/Group_cohomology" title="Group cohomology">group cohomology</a> and <a href="/wiki/Group_homology" class="mw-redirect" title="Group homology">group homology</a>, which use the same superscript/subscript convention. <div class="mw-heading mw-heading3"><h3 id="Invariant_subsets">Invariant subsets</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=12" title="Edit section: Invariant subsets">edit</a>]</div> If Y is a <a href="/wiki/Subset" title="Subset">subset</a> of X, then G⋅Y denotes the set {g⋅y : g ∈ G and y ∈ Y}. The subset Y is said to be invariant under G if G⋅Y = Y (which is equivalent G⋅Y ⊆ Y). In that case, G also operates on Y by <a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">restricting</a> the action to Y. The subset Y is called fixed under G if g⋅y = y for all g in G and all y in Y. Every subset that is fixed under G is also invariant under G, but not conversely. Every orbit is an invariant subset of X on which G acts <a href="/wiki/Group_action_(mathematics)#Notable_properties_of_actions" class="mw-redirect" title="Group action (mathematics)">transitively</a>. Conversely, any invariant subset of X is a union of orbits. The action of G on X is transitive if and only if all elements are equivalent, meaning that there is only one orbit. A G-invariant element of X is x ∈ X such that g⋅x = x for all g ∈ G. The set of all such x is denoted XG and called the G-invariants of X. When X is a <a href="/wiki/G-module" title="G-module">G-module</a>, XG is the zeroth <a href="/wiki/Group_cohomology" title="Group cohomology">cohomology</a> group of G with coefficients in X, and the higher cohomology groups are the <a href="/wiki/Derived_functor" title="Derived functor">derived functors</a> of the <a href="/wiki/Functor" title="Functor">functor</a> of G-invariants. <div class="mw-heading mw-heading3"><h3 id="Fixed_points_and_stabilizer_subgroups">Fixed points and stabilizer subgroups</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=13" title="Edit section: Fixed points and stabilizer subgroups">edit</a>]</div> Given g in G and x in X with g⋅x = x, it is said that "x is a fixed point of g" or that "g fixes x". For every x in X, the <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">stabilizer subgroup of G with respect to x (also called the isotropy group or little group<a href="#cite_note-Procesi-10">[10]</a>) is the set of all elements in G that fix x: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo>:</mo> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a84445f34b016a3662405602af20a618e4ad1861" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.311ex; height:2.843ex;" alt="{\displaystyle G_{x}=\{g\in G:g{\cdot }x=x\}.}"> This is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of G, though typically not a normal one. The action of G on X is <a href="/wiki/Group_action_(mathematics)#Notable_properties_of_actions" class="mw-redirect" title="Group action (mathematics)">free</a> if and only if all stabilizers are trivial. The kernel N of the homomorphism with the symmetric group, G → Sym(X), is given by the <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of the stabilizers Gx for all x in X. If N is trivial, the action is said to be faithful (or effective). Let x and y be two elements in X, and let g be a group element such that y = g⋅x. Then the two stabilizer groups Gx and Gy are related by Gy = gGxg−1. Proof: by definition, h ∈ Gy if and only if h⋅(g⋅x) = g⋅x. Applying g−1 to both sides of this equality yields (g−1hg)⋅x = x; that is, g−1hg ∈ Gx. An opposite inclusion follows similarly by taking h ∈ Gx and x = g−1⋅y. The above says that the stabilizers of elements in the same orbit are <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugate</a> to each other. Thus, to each orbit, we can associate a <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy class</a> of a subgroup of G (that is, the set of all conjugates of the subgroup). Let (H) denote the conjugacy class of H. Then the orbit O has type (H) if the stabilizer Gx of some/any x in O belongs to (H). A maximal orbit type is often called a <a href="/wiki/Principal_orbit_type" class="mw-redirect" title="Principal orbit type">principal orbit type</a>. <div class="mw-heading mw-heading3"><h3 id="Orbit-stabilizer_theorem"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">Orbit-stabilizer theorem</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=14" title="Edit section: Orbit-stabilizer theorem">edit</a>]</div> Orbits and stabilizers are closely related. For a fixed x in X, consider the map f : G → X given by g ↦ g⋅x. By definition the image f(G) of this map is the orbit G⋅x. The condition for two elements to have the same image is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>h</mi> <mo>∈</mo> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>h</mi> <mo>∈</mo> <mi>g</mi> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2abb119ece1b4cabc0f87b562bc3441c50c8f755" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.591ex; height:3.176ex;" alt="{\displaystyle f(g)=f(h)\iff g{\cdot }x=h{\cdot }x\iff g^{-1}h{\cdot }x=x\iff g^{-1}h\in G_{x}\iff h\in gG_{x}.}"> In other words, f(g) = f(h) if and only if g and h lie in the same <a href="/wiki/Coset" title="Coset">coset</a> for the stabilizer subgroup Gx. Thus, the <a href="/wiki/Fiber_(mathematics)" title="Fiber (mathematics)">fiber</a> f−1({y}) of f over any y in G⋅x is contained in such a coset, and every such coset also occurs as a fiber. Therefore f induces a bijection between the set G / Gx of cosets for the stabilizer subgroup and the orbit G⋅x, which sends gGx ↦ g⋅x.<a href="#cite_note-11">[11]</a> This result is known as the orbit-stabilizer theorem. If G is finite then the orbit-stabilizer theorem, together with <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a>, gives <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mo>⋅</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>G</mi> <mspace width="thinmathspace" /> <mo>:</mo> <mspace width="thinmathspace" /> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2cf9bb1f43b44e6798feefb9854929e9dcdc33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.38ex; height:2.843ex;" alt="{\displaystyle |G\cdot x|=[G\,:\,G_{x}]=|G|/|G_{x}|,}"> in other words the length of the orbit of x times the order of its stabilizer is the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order of the group</a>. In particular that implies that the orbit length is a divisor of the group order. <dl><dd>Example: Let G be a group of prime order p acting on a set X with k elements. Since each orbit has either 1 or p elements, there are at least k mod p orbits of length 1 which are G-invariant elements. More specifically, k and the number of G-invariant elements are congruent modulo p.<a href="#cite_note-12">[12]</a></dd></dl> This result is especially useful since it can be employed for counting arguments (typically in situations where X is finite as well). <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Labeled_cube_graph.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Labeled_cube_graph.png/220px-Labeled_cube_graph.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Labeled_cube_graph.png/330px-Labeled_cube_graph.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/a4/Labeled_cube_graph.png 2x" data-file-width="400" data-file-height="360" /></a><figcaption>Cubical graph with vertices labeled</figcaption></figure> <dl><dd>Example: We can use the orbit-stabilizer theorem to count the automorphisms of a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>. Consider the <a href="/wiki/Cubical_graph" class="mw-redirect" title="Cubical graph">cubical graph</a> as pictured, and let G denote its <a href="/wiki/Graph_automorphism" title="Graph automorphism">automorphism</a> group. Then G acts on the set of vertices {1, 2, ..., 8}, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, |G| = |G ⋅ 1| |G1| = 8 |G1|. Applying the theorem now to the stabilizer G1, we can obtain |G1| = |(G1) ⋅ 2| |(G1)2|. Any element of G that fixes 1 must send 2 to either 2, 4, or 5. As an example of such automorphisms consider the rotation around the diagonal axis through 1 and 7 by 2π/3, which permutes 2, 4, 5 and 3, 6, 8, and fixes 1 and 7. Thus, |(G1) ⋅ 2| = 3. Applying the theorem a third time gives |(G1)2| = |((G1)2) ⋅ 3| |((G1)2)3|. Any element of G that fixes 1 and 2 must send 3 to either 3 or 6. Reflecting the cube at the plane through 1, 2, 7 and 8 is such an automorphism sending 3 to 6, thus |((G1)2) ⋅ 3| = 2. One also sees that ((G1)2)3 consists only of the identity automorphism, as any element of G fixing 1, 2 and 3 must also fix all other vertices, since they are determined by their adjacency to 1, 2 and 3. Combining the preceding calculations, we can now obtain |G| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Burnside's_lemma">Burnside's lemma</h3>[<a href="/w/index.php?title=Group_action&action=edit&section=15" title="Edit section: Burnside's lemma">edit</a>]</div> A result closely related to the orbit-stabilizer theorem is <a href="/wiki/Burnside%27s_lemma" title="Burnside's lemma">Burnside's lemma</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e179e15955324945461f18a84260256e5deb0d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.406ex; height:6.676ex;" alt="{\displaystyle |X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|,}"> where Xg is the set of points fixed by g. This result is mainly of use when G and X are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element. Fixing a group G, the set of formal differences of finite G-sets forms a ring called the <a href="/wiki/Burnside_ring" title="Burnside ring">Burnside ring</a> of G, where addition corresponds to <a href="/wiki/Disjoint_union" title="Disjoint union">disjoint union</a>, and multiplication to <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a>. <div class="mw-heading mw-heading2"><h2 id="Examples_2">Examples</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=16" title="Edit section: Examples">edit</a>]</div> <ul><li>The <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">trivial action of any group G on any set X is defined by g⋅x = x for all g in G and all x in X; that is, every group element induces the <a href="/wiki/Identity_function" title="Identity function">identity permutation</a> on X.<a href="#cite_note-13">[13]</a></li> <li>In every group G, left multiplication is an action of G on G: g⋅x = gx for all g, x in G. This action is free and transitive (regular), and forms the basis of a rapid proof of <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a> – that every group is isomorphic to a subgroup of the symmetric group of permutations of the set G.</li> <li>In every group G with subgroup H, left multiplication is an action of G on the set of cosets G / H: g⋅aH = gaH for all g, a in G. In particular if H contains no nontrivial <a href="/wiki/Normal_subgroups" class="mw-redirect" title="Normal subgroups">normal subgroups</a> of G this induces an isomorphism from G to a subgroup of the permutation group of <a href="/wiki/Degree_of_a_permutation_group" class="mw-redirect" title="Degree of a permutation group">degree</a> [G : H].</li> <li>In every group G, <a href="/wiki/Inner_automorphism" title="Inner automorphism">conjugation</a> is an action of G on G: g⋅x = gxg−1. An exponential notation is commonly used for the right-action variant: xg = g−1xg; it satisfies (xg)h = xgh.</li> <li>In every group G with subgroup H, conjugation is an action of G on conjugates of H: g⋅K = gKg−1 for all g in G and K conjugates of H.</li> <li>An action of Z on a set X uniquely determines and is determined by an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> of X, given by the action of 1. Similarly, an action of Z / 2Z on X is equivalent to the data of an <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> of X.</li> <li>The symmetric group Sn and its subgroups act on the set {1, ..., n} by permuting its elements</li> <li>The <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of a polyhedron acts on the set of vertices of that polyhedron. It also acts on the set of faces or the set of edges of the polyhedron.</li> <li>The symmetry group of any geometrical object acts on the set of points of that object.</li> <li>For a <a href="/wiki/Coordinate_space" class="mw-redirect" title="Coordinate space">coordinate space</a> V over a field F with group of units F*, the mapping F* × V → V given by a × (x1, x2, ..., xn) ↦ (ax1, ax2, ..., axn) is a group action called <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>.</li> <li>The automorphism group of a vector space (or <a href="/wiki/Graph_theory" title="Graph theory">graph</a>, or group, or ring ...) acts on the vector space (or set of vertices of the graph, or group, or ring ...).</li> <li>The general linear group GL(n, K) and its subgroups, particularly its <a href="/wiki/Lie_subgroup" class="mw-redirect" title="Lie subgroup">Lie subgroups</a> (including the special linear group SL(n, K), <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(n, K), special orthogonal group SO(n, K), and <a href="/wiki/Symplectic_group" title="Symplectic group">symplectic group</a> Sp(n, K)) are <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> that act on the vector space Kn. The group operations are given by multiplying the matrices from the groups with the vectors from Kn.</li> <li>The general linear group GL(n, Z) acts on Zn by natural matrix action. The orbits of its action are classified by the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of coordinates of the vector in Zn.</li> <li>The <a href="/wiki/Affine_group" title="Affine group">affine group</a> acts <a href="#Notable_properties_of_actions">transitively</a> on the points of an <a href="/wiki/Affine_space" title="Affine space">affine space</a>, and the subgroup V of the affine group (that is, a vector space) has transitive and free (that is, regular) action on these points;<a href="#cite_note-14">[14]</a> indeed this can be used to give a definition of an <a href="/wiki/Affine_space#Definition" title="Affine space">affine space</a>.</li> <li>The <a href="/wiki/Projective_linear_group" title="Projective linear group">projective linear group</a> PGL(n + 1, K) and its subgroups, particularly its Lie subgroups, which are Lie groups that act on the <a href="/wiki/Projective_space" title="Projective space">projective space</a> Pn(K). This is a quotient of the action of the general linear group on projective space. Particularly notable is PGL(2, K), the symmetries of the projective line, which is sharply 3-transitive, preserving the <a href="/wiki/Cross_ratio" class="mw-redirect" title="Cross ratio">cross ratio</a>; the <a href="/wiki/M%C3%B6bius_group" class="mw-redirect" title="Möbius group">Möbius group</a> PGL(2, C) is of particular interest.</li> <li>The <a href="/wiki/Isometry" title="Isometry">isometries</a> of the plane act on the set of 2D images and patterns, such as <a href="/wiki/Wallpaper_group" title="Wallpaper group">wallpaper patterns</a>. The definition can be made more precise by specifying what is meant by image or pattern, for example, a function of position with values in a set of colors. Isometries are in fact one example of affine group (action).[<a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute">dubious</a> – <a href="/wiki/Talk:Group_action#Dubious" title="Talk:Group action">discuss</a>]</li> <li>The sets acted on by a group G comprise the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of G-sets in which the objects are G-sets and the <a href="/wiki/Morphism" title="Morphism">morphisms</a> are G-set homomorphisms: functions f : X → Y such that g⋅(f(x)) = f(g⋅x) for every g in G.</li> <li>The <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of a <a href="/wiki/Field_extension" title="Field extension">field extension</a> L / K acts on the field L but has only a trivial action on elements of the subfield K. Subgroups of Gal(L / K) correspond to subfields of L that contain K, that is, intermediate field extensions between L and K.</li> <li>The additive group of the <a href="/wiki/Real_number" title="Real number">real numbers</a> (R, +) acts on the <a href="/wiki/Phase_space" title="Phase space">phase space</a> of "<a href="/wiki/Well-behaved" class="mw-redirect" title="Well-behaved">well-behaved</a>" systems in <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> (and in more general <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>) by <a href="/wiki/Time_translation" class="mw-redirect" title="Time translation">time translation</a>: if t is in R and x is in the phase space, then x describes a state of the system, and t + x is defined to be the state of the system t seconds later if t is positive or −t seconds ago if t is negative.</li> <li>The additive group of the real numbers (R, +) acts on the set of real functions of a real variable in various ways, with (t⋅f)(x) equal to, for example, f(x + t), f(x) + t, f(xet), f(x)et, f(x + t)et, or f(xet) + t, but not f(xet + t).</li> <li>Given a group action of G on X, we can define an induced action of G on the <a href="/wiki/Power_set" title="Power set">power set</a> of X, by setting g⋅U = {g⋅u : u ∈ U} for every subset U of X and every g in G. This is useful, for instance, in studying the action of the large <a href="/wiki/Mathieu_group" title="Mathieu group">Mathieu group</a> on a 24-set and in studying symmetry in certain models of <a href="/wiki/Finite_geometry" title="Finite geometry">finite geometries</a>.</li> <li>The <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> with <a href="/wiki/Norm_of_a_quaternion" class="mw-redirect" title="Norm of a quaternion">norm</a> 1 (the <a href="/wiki/Versor" title="Versor">versors</a>), as a multiplicative group, act on R3: for any such quaternion z = cos α/2 + v sin α/2, the mapping f(x) = zxz* is a counterclockwise rotation through an angle α about an axis given by a unit vector v; z is the same rotation; see <a href="/wiki/Quaternions_and_spatial_rotation" title="Quaternions and spatial rotation">quaternions and spatial rotation</a>. This is not a faithful action because the quaternion −1 leaves all points where they were, as does the quaternion 1.</li> <li>Given left G-sets X, Y, there is a left G-set YX whose elements are G-equivariant maps α : X × G → Y, and with left G-action given by g⋅α = α ∘ (idX × –g) (where "–g" indicates right multiplication by g). This G-set has the property that its fixed points correspond to equivariant maps X → Y; more generally, it is an <a href="/wiki/Exponential_object" title="Exponential object">exponential object</a> in the category of G-sets.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Group_actions_and_groupoids">Group actions and groupoids</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=17" title="Edit section: Group actions and groupoids">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/Groupoid#Group_action" title="Groupoid">Groupoid § Group action</a></div> The notion of group action can be encoded by the action <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> G′ = G ⋉ X associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits of the action are its components. <div class="mw-heading mw-heading2"><h2 id="Morphisms_and_isomorphisms_between_G-sets">Morphisms and isomorphisms between G-sets</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=18" title="Edit section: Morphisms and isomorphisms between G-sets">edit</a>]</div> If X and Y are two G-sets, a morphism from X to Y is a function f : X → Y such that f(g⋅x) = g⋅f(x) for all g in G and all x in X. Morphisms of G-sets are also called <a href="/wiki/Equivariant_map" title="Equivariant map">equivariant maps</a> or G-maps. The composition of two morphisms is again a morphism. If a morphism f is bijective, then its inverse is also a morphism. In this case f is called an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>, and the two G-sets X and Y are called isomorphic; for all practical purposes, isomorphic G-sets are indistinguishable. Some example isomorphisms: <ul><li>Every regular G action is isomorphic to the action of G on G given by left multiplication.</li> <li>Every free G action is isomorphic to G × S, where S is some set and G acts on G × S by left multiplication on the first coordinate. (S can be taken to be the set of orbits X / G.)</li> <li>Every transitive G action is isomorphic to left multiplication by G on the set of left cosets of some subgroup H of G. (H can be taken to be the stabilizer group of any element of the original G-set.)</li></ul> With this notion of morphism, the collection of all G-sets forms a <a href="/wiki/Category_theory" title="Category theory">category</a>; this category is a <a href="/wiki/Grothendieck_topos" class="mw-redirect" title="Grothendieck topos">Grothendieck topos</a> (in fact, assuming a classical <a href="/wiki/Metalogic" title="Metalogic">metalogic</a>, this <a href="/wiki/Topos" title="Topos">topos</a> will even be Boolean). <div class="mw-heading mw-heading2"><h2 id="Variants_and_generalizations">Variants and generalizations</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=19" title="Edit section: Variants and generalizations">edit</a>]</div> We can also consider actions of <a href="/wiki/Monoid" title="Monoid">monoids</a> on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See <a href="/wiki/Semigroup_action" title="Semigroup action">semigroup action</a>. Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object X of some category, and then define an action on X as a monoid homomorphism into the monoid of <a href="/wiki/Endomorphisms" class="mw-redirect" title="Endomorphisms">endomorphisms</a> of X. If X has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain <a href="/wiki/Group_representation" title="Group representation">group representations</a> in this fashion. We can view a group G as a category with a single object in which every morphism is <a href="/wiki/Inverse_element" title="Inverse element">invertible</a>.<a href="#cite_note-15">[15]</a> A (left) group action is then nothing but a (covariant) <a href="/wiki/Functor" title="Functor">functor</a> from G to the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a>, and a group representation is a functor from G to the <a href="/wiki/Category_of_vector_spaces" class="mw-redirect" title="Category of vector spaces">category of vector spaces</a>.<a href="#cite_note-16">[16]</a> A morphism between G-sets is then a <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformation</a> between the group action functors.<a href="#cite_note-17">[17]</a> In analogy, an action of a <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> is a functor from the groupoid to the category of sets or to some other category. In addition to <a href="/wiki/Continuous_group_action" title="Continuous group action">continuous actions</a> of topological groups on topological spaces, one also often considers <a href="/wiki/Lie_group_action" title="Lie group action">smooth actions</a> of Lie groups on <a href="/wiki/Manifold" title="Manifold">smooth manifolds</a>, regular actions of <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a> on <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a>, and <a href="/wiki/Group-scheme_action" title="Group-scheme action">actions</a> of <a href="/wiki/Group_scheme" title="Group scheme">group schemes</a> on <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a>. All of these are examples of <a href="/wiki/Group_object" title="Group object">group objects</a> acting on objects of their respective category. <div class="mw-heading mw-heading2"><h2 id="Gallery">Gallery</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=20" title="Edit section: Gallery">edit</a>]</div> <ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 210px;"><a href="/wiki/File:Octahedral-group-action.png" class="mw-file-description" title="Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group."><img alt="Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Octahedral-group-action.png/200px-Octahedral-group-action.png" decoding="async" width="200" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Octahedral-group-action.png/300px-Octahedral-group-action.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Octahedral-group-action.png/400px-Octahedral-group-action.png 2x" data-file-width="1280" data-file-height="1024" /></a></div> <div class="gallerytext">Orbit of a fundamental spherical triangle (marked in red) under action of the full octahedral group.</div> </li> <li class="gallerybox" style="width: 235px"> <div class="thumb" style="width: 230px; height: 210px;"><a href="/wiki/File:Icosahedral-group-action.png" class="mw-file-description" title="Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group."><img alt="Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group." src="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Icosahedral-group-action.png/200px-Icosahedral-group-action.png" decoding="async" width="200" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Icosahedral-group-action.png/300px-Icosahedral-group-action.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/59/Icosahedral-group-action.png/400px-Icosahedral-group-action.png 2x" data-file-width="1280" data-file-height="1024" /></a></div> <div class="gallerytext">Orbit of a fundamental spherical triangle (marked in red) under action of the full icosahedral group.</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=21" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Gain_graph" title="Gain graph">Gain graph</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Measurable_group_action" class="mw-redirect" title="Measurable group action">Measurable group action</a></li> <li><a href="/wiki/Monoid_action" class="mw-redirect" title="Monoid action">Monoid action</a></li> <li><a href="/wiki/Young%E2%80%93Deruyts_development" title="Young–Deruyts development">Young–Deruyts development</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=22" 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> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=23" 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-1"><a href="#cite_ref-1">^</a> <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="CITEREFEie_&_Chang2010" class="citation book cs1">Eie & Chang (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jozIZ0qrkk8C&pg=PA144&dq=%22group+action%22">A Course on Abstract Algebra</a>. p. 144.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+on+Abstract+Algebra&rft.pages=144&rft.date=2010&rft.au=Eie+%26+Chang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjozIZ0qrkk8C%26pg%3DPA144%26dq%3D%2522group%2Baction%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> This is done, for example, by <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2008" class="citation book cs1">Smith (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PQUAQh04lrUC&pg=PA253&dq=%22group+action%22">Introduction to abstract algebra</a>. p. 253.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+abstract+algebra&rft.pages=253&rft.date=2008&rft.au=Smith&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DPQUAQh04lrUC%26pg%3DPA253%26dq%3D%2522group%2Baction%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Right_Group_Action_Axioms">"Definition:Right Group Action Axioms"</a>. Proof Wiki. Retrieved 19 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Proof+Wiki&rft.atitle=Definition%3ARight+Group+Action+Axioms&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3ARight_Group_Action_Axioms&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-FOOTNOTEThurston1997Definition_3.5.1(iv)-4"><a href="#cite_ref-FOOTNOTEThurston1997Definition_3.5.1(iv)_4-0">^</a> <a href="#CITEREFThurston1997">Thurston 1997</a>, Definition 3.5.1(iv). </li> <li id="cite_note-FOOTNOTEKapovich2009p._73-5"><a href="#cite_ref-FOOTNOTEKapovich2009p._73_5-0">^</a> <a href="#CITEREFKapovich2009">Kapovich 2009</a>, p. 73. </li> <li id="cite_note-FOOTNOTEThurston1980176-6"><a href="#cite_ref-FOOTNOTEThurston1980176_6-0">^</a> <a href="#CITEREFThurston1980">Thurston 1980</a>, p. 176. </li> <li id="cite_note-FOOTNOTEHatcher2002p._72-7"><a href="#cite_ref-FOOTNOTEHatcher2002p._72_7-0">^</a> <a href="#CITEREFHatcher2002">Hatcher 2002</a>, p. 72. </li> <li id="cite_note-FOOTNOTEMaskit1988II.A.1,_II.A.2-8"><a href="#cite_ref-FOOTNOTEMaskit1988II.A.1,_II.A.2_8-0">^</a> <a href="#CITEREFMaskit1988">Maskit 1988</a>, II.A.1, II.A.2. </li> <li id="cite_note-FOOTNOTEtom_Dieck1987-9"><a href="#cite_ref-FOOTNOTEtom_Dieck1987_9-0">^</a> <a href="#CITEREFtom_Dieck1987">tom Dieck 1987</a>. </li> <li id="cite_note-Procesi-10"><a href="#cite_ref-Procesi_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProcesi2007" class="citation book cs1">Procesi, Claudio (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Sl8OAGYRz_AC&q=%22little+group%22+action&pg=PA5">Lie Groups: An Approach through Invariants and Representations</a>. Springer Science & Business Media. p. 5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387289298" title="Special:BookSources/9780387289298"><bdi>9780387289298</bdi></a>. Retrieved 23 February 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+Groups%3A+An+Approach+through+Invariants+and+Representations&rft.pages=5&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=9780387289298&rft.aulast=Procesi&rft.aufirst=Claudio&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSl8OAGYRz_AC%26q%3D%2522little%2Bgroup%2522%2Baction%26pg%3DPA5&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> M. Artin, Algebra, Proposition 6.8.4 on p. 179 </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarter2009" class="citation book cs1">Carter, Nathan (2009). Visual Group Theory (1st ed.). The Mathematical Association of America. p. 200. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0883857571" title="Special:BookSources/978-0883857571"><bdi>978-0883857571</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Visual+Group+Theory&rft.pages=200&rft.edition=1st&rft.pub=The+Mathematical+Association+of+America&rft.date=2009&rft.isbn=978-0883857571&rft.aulast=Carter&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEie_&_Chang2010" class="citation book cs1">Eie & Chang (2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jozIZ0qrkk8C&pg=PA144&dq=%22trivial+action%22">A Course on Abstract Algebra</a>. p. 145.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+on+Abstract+Algebra&rft.pages=145&rft.date=2010&rft.au=Eie+%26+Chang&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjozIZ0qrkk8C%26pg%3DPA144%26dq%3D%2522trivial%2Baction%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReid2005" class="citation book cs1">Reid, Miles (2005). Geometry and topology. Cambridge, UK New York: Cambridge University Press. p. 170. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521613255" title="Special:BookSources/9780521613255"><bdi>9780521613255</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometry+and+topology&rft.place=Cambridge%2C+UK+New+York&rft.pages=170&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=9780521613255&rft.aulast=Reid&rft.aufirst=Miles&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFPerrone2024">Perrone (2024)</a>, pp. 7–9 </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFPerrone2024">Perrone (2024)</a>, pp. 36–39 </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFPerrone2024">Perrone (2024)</a>, pp. 69–71 </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=24" title="Edit section: References">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAschbacher2000" class="citation book cs1"><a href="/wiki/Michael_Aschbacher" title="Michael Aschbacher">Aschbacher, Michael</a> (2000). Finite Group Theory. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-78675-1" title="Special:BookSources/978-0-521-78675-1"><bdi>978-0-521-78675-1</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=1777008">1777008</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Group+Theory&rft.pub=Cambridge+University+Press&rft.date=2000&rft.isbn=978-0-521-78675-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1777008%23id-name%3DMR&rft.aulast=Aschbacher&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDummitRichard_Foote2003" class="citation book cs1">Dummit, David; Richard Foote (2003). Abstract Algebra (3rd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-43334-9" title="Special:BookSources/0-471-43334-9"><bdi>0-471-43334-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra&rft.edition=3rd&rft.pub=Wiley&rft.date=2003&rft.isbn=0-471-43334-9&rft.aulast=Dummit&rft.aufirst=David&rft.au=Richard+Foote&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEieChang2010" class="citation book cs1">Eie, Minking; Chang, Shou-Te (2010). A Course on Abstract Algebra. World Scientific. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4271-88-2" title="Special:BookSources/978-981-4271-88-2"><bdi>978-981-4271-88-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+on+Abstract+Algebra&rft.pub=World+Scientific&rft.date=2010&rft.isbn=978-981-4271-88-2&rft.aulast=Eie&rft.aufirst=Minking&rft.au=Chang%2C+Shou-Te&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" 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>, Cambridge University Press, <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>, <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=1867354">1867354</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_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1867354%23id-name%3DMR&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+action" class="Z3988">.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman1995" class="citation book cs1">Rotman, Joseph (1995). An Introduction to the Theory of Groups. Graduate Texts in Mathematics 148 (4th ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94285-8" title="Special:BookSources/0-387-94285-8"><bdi>0-387-94285-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Groups&rft.edition=4th&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=0-387-94285-8&rft.aulast=Rotman&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSmith2008" class="citation book cs1">Smith, Jonathan D.H. (2008). Introduction to abstract algebra. Textbooks in mathematics. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4200-6371-4" title="Special:BookSources/978-1-4200-6371-4"><bdi>978-1-4200-6371-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+abstract+algebra&rft.series=Textbooks+in+mathematics&rft.pub=CRC+Press&rft.date=2008&rft.isbn=978-1-4200-6371-4&rft.aulast=Smith&rft.aufirst=Jonathan+D.H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKapovich2009" class="citation cs2">Kapovich, Michael (2009), Hyperbolic manifolds and discrete groups, Modern Birkhäuser Classics, Birkhäuser, pp. xxvii+467, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8176-4912-8" title="Special:BookSources/978-0-8176-4912-8"><bdi>978-0-8176-4912-8</bdi></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:1180.57001">1180.57001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hyperbolic+manifolds+and+discrete+groups&rft.series=Modern+Birkh%C3%A4user+Classics&rft.pages=xxvii%2B467&rft.pub=Birkh%C3%A4user&rft.date=2009&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1180.57001%23id-name%3DZbl&rft.isbn=978-0-8176-4912-8&rft.aulast=Kapovich&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMaskit1988" class="citation cs2">Maskit, Bernard (1988), Kleinian groups, Grundlehren der Mathematischen Wissenschaften, vol. 287, Springer-Verlag, pp. XIII+326, <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:0627.30039">0627.30039</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Kleinian+groups&rft.series=Grundlehren+der+Mathematischen+Wissenschaften&rft.pages=XIII%2B326&rft.pub=Springer-Verlag&rft.date=1988&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0627.30039%23id-name%3DZbl&rft.aulast=Maskit&rft.aufirst=Bernard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPerrone2024" class="citation cs2">Perrone, Paolo (2024), Starting Category Theory, World Scientific, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F9789811286018_0005">10.1142/9789811286018_0005</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-12-8600-1" title="Special:BookSources/978-981-12-8600-1"><bdi>978-981-12-8600-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Starting+Category+Theory&rft.pub=World+Scientific&rft.date=2024&rft_id=info%3Adoi%2F10.1142%2F9789811286018_0005&rft.isbn=978-981-12-8600-1&rft.aulast=Perrone&rft.aufirst=Paolo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThurston1980" class="citation cs2">Thurston, William (1980), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20200727020107/http://library.msri.org/books/gt3m/">The geometry and topology of three-manifolds</a>, Princeton lecture notes, p. 175, archived from <a rel="nofollow" class="external text" href="http://library.msri.org/books/gt3m/">the original</a> on 2020-07-27, retrieved 2016-02-08</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+geometry+and+topology+of+three-manifolds&rft.series=Princeton+lecture+notes&rft.pages=175&rft.date=1980&rft.aulast=Thurston&rft.aufirst=William&rft_id=http%3A%2F%2Flibrary.msri.org%2Fbooks%2Fgt3m%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThurston1997" class="citation cs2">Thurston, William P. (1997), Three-dimensional geometry and topology. Vol. 1., Princeton Mathematical Series, vol. 35, Princeton University Press, pp. x+311, <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:0873.57001">0873.57001</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Three-dimensional+geometry+and+topology.+Vol.+1.&rft.series=Princeton+Mathematical+Series&rft.pages=x%2B311&rft.pub=Princeton+University+Press&rft.date=1997&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0873.57001%23id-name%3DZbl&rft.aulast=Thurston&rft.aufirst=William+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFtom_Dieck1987" class="citation cs2">tom Dieck, Tammo (1987), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=azcQhi6XeioC">Transformation groups</a>, de Gruyter Studies in Mathematics, vol. 8, Berlin: Walter de Gruyter & Co., p. 29, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9783110858372.312">10.1515/9783110858372.312</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-11-009745-0" title="Special:BookSources/978-3-11-009745-0"><bdi>978-3-11-009745-0</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=0889050">0889050</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transformation+groups&rft.place=Berlin&rft.series=de+Gruyter+Studies+in+Mathematics&rft.pages=29&rft.pub=Walter+de+Gruyter+%26+Co.&rft.date=1987&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D889050%23id-name%3DMR&rft_id=info%3Adoi%2F10.1515%2F9783110858372.312&rft.isbn=978-3-11-009745-0&rft.aulast=tom+Dieck&rft.aufirst=Tammo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DazcQhi6XeioC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Group_action&action=edit&section=25" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Action_of_a_group_on_a_manifold">"Action of a group on a manifold"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Action+of+a+group+on+a+manifold&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAction_of_a_group_on_a_manifold&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><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/GroupAction.html">"Group Action"</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+Action&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGroupAction.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+action" 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 authority-control" aria-label="Navbox" 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/Q288465#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://id.loc.gov/authorities/sh85057471">United States</a></li><li><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007543475605171">Israel</a></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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_action&oldid=1256994680#Orbits_and_stabilizers">https://en.wikipedia.org/w/index.php?title=Group_action&oldid=1256994680#Orbits_and_stabilizers</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:Group_actions_(mathematics)" title="Category:Group actions (mathematics)">Group actions (mathematics)</a></li><li><a href="/wiki/Category:Representation_theory_of_groups" title="Category:Representation theory of groups">Representation theory of groups</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:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_May_2023" title="Category:Articles with unsourced statements from May 2023">Articles with unsourced statements from May 2023</a></li><li><a href="/wiki/Category:All_accuracy_disputes" title="Category:All accuracy disputes">All accuracy disputes</a></li><li><a href="/wiki/Category:Articles_with_disputed_statements_from_March_2015" title="Category:Articles with disputed statements from March 2015">Articles with disputed statements from March 2015</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 November 2024, at 16:58 (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_action&mobileaction=toggle_view_mobile#Orbits_and_stabilizers" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></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"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-57488d5c7d-sj6fv","wgBackendResponseTime":175,"wgPageParseReport":{"limitreport":{"cputime":"1.376","walltime":"1.679","ppvisitednodes":{"value":27115,"limit":1000000},"postexpandincludesize":{"value":172518,"limit":2097152},"templateargumentsize":{"value":43605,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":8,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":114562,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 1369.904 1 -total"," 40.45% 554.132 503 Template:Math"," 12.41% 170.070 2 Template:Reflist"," 12.04% 164.937 11 Template:Cite_book"," 8.53% 116.822 1 Template:Group_theory_sidebar"," 8.37% 114.635 1 Template:Sidebar_with_collapsible_lists"," 7.43% 101.760 516 Template:Main_other"," 6.22% 85.199 1 Template:Short_description"," 6.21% 85.112 1 Template:Citation_needed"," 4.47% 61.287 1 Template:Authority_control"]},"scribunto":{"limitreport-timeusage":{"value":"0.689","limit":"10.000"},"limitreport-memusage":{"value":7633863,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFAschbacher2000\"] = 1,\n [\"CITEREFCarter2009\"] = 1,\n [\"CITEREFDummitRichard_Foote2003\"] = 1,\n [\"CITEREFEieChang2010\"] = 1,\n [\"CITEREFEie_\u0026amp;_Chang2010\"] = 2,\n [\"CITEREFHatcher2002\"] = 1,\n [\"CITEREFKapovich2009\"] = 1,\n [\"CITEREFMaskit1988\"] = 1,\n [\"CITEREFPerrone2024\"] = 1,\n [\"CITEREFProcesi2007\"] = 1,\n [\"CITEREFReid2005\"] = 1,\n [\"CITEREFRotman1995\"] = 1,\n [\"CITEREFSmith2008\"] = 2,\n [\"CITEREFThurston1980\"] = 1,\n [\"CITEREFThurston1997\"] = 1,\n [\"CITEREFtom_Dieck1987\"] = 1,\n}\ntemplate_list = table#1 {\n [\"About\"] = 1,\n [\"Abs\"] = 13,\n [\"Authority control\"] = 1,\n [\"Citation\"] = 7,\n [\"Citation needed\"] = 1,\n [\"Cite book\"] = 11,\n [\"Cite web\"] = 1,\n [\"Dubious\"] = 1,\n [\"Em\"] = 4,\n [\"Google books\"] = 3,\n [\"Group theory sidebar\"] = 1,\n [\"Harvp\"] = 3,\n [\"I sup\"] = 2,\n [\"Main\"] = 4,\n [\"Math\"] = 503,\n [\"MathWorld\"] = 1,\n [\"Mset\"] = 8,\n [\"Mvar\"] = 41,\n [\"Notelist\"] = 1,\n [\"Reflist\"] = 1,\n [\"Sfn\"] = 6,\n [\"Short description\"] = 1,\n [\"Springer\"] = 1,\n [\"Sub\"] = 1,\n [\"Visible anchor\"] = 16,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.canary-54459f6c5c-fw256","timestamp":"20241127005307","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Group action","url":"https:\/\/en.wikipedia.org\/wiki\/Group_action#Orbits_and_stabilizers","sameAs":"http:\/\/www.wikidata.org\/entity\/Q288465","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q288465","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":"2001-11-02T15:49:34Z","dateModified":"2024-11-12T16:58:18Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/5\/5f\/Cyclic_group.svg","headline":"operation of the elements of a group as transformations or automorphisms (mathematics)"}</script> </body> </html>