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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-Subgroup_tests" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subgroup_tests"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Subgroup tests</span> </div> </a> <ul id="toc-Subgroup_tests-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_properties_of_subgroups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_properties_of_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Basic properties of subgroups</span> </div> </a> <ul id="toc-Basic_properties_of_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cosets_and_Lagrange's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cosets_and_Lagrange's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Cosets and Lagrange's theorem</span> </div> </a> <ul id="toc-Cosets_and_Lagrange's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_Subgroups_of_Z8" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example:_Subgroups_of_Z8"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Example: Subgroups of Z<sub>8</sub></span> </div> </a> <ul id="toc-Example:_Subgroups_of_Z8-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example:_Subgroups_of_S4" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Example:_Subgroups_of_S4"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Example: Subgroups of S<sub>4</sub></span> </div> </a> <button aria-controls="toc-Example:_Subgroups_of_S4-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Example: Subgroups of S<sub>4</sub> subsection</span> </button> <ul id="toc-Example:_Subgroups_of_S4-sublist" class="vector-toc-list"> <li id="toc-24_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#24_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>24 elements</span> </div> </a> <ul id="toc-24_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-12_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#12_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>12 elements</span> </div> </a> <ul id="toc-12_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-8_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#8_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>8 elements</span> </div> </a> <ul id="toc-8_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-6_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#6_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>6 elements</span> </div> </a> <ul id="toc-6_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-4_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#4_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>4 elements</span> </div> </a> <ul id="toc-4_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-3_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#3_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>3 elements</span> </div> </a> <ul id="toc-3_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#2_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>2 elements</span> </div> </a> <ul id="toc-2_elements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-1_element" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#1_element"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>1 element</span> </div> </a> <ul id="toc-1_element-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Other examples</span> </div> </a> <ul id="toc-Other_examples-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"> <span class="vector-toc-numb">7</span> <span>See also</span> </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"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">Subgroup</span></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 40 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-40" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D8%AC%D8%B2%D8%A6%D9%8A%D8%A9" title="زمرة جزئية – Arabic" lang="ar" hreflang="ar" data-title="زمرة جزئية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Altqrup" title="Altqrup – Azerbaijani" lang="az" hreflang="az" data-title="Altqrup" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Падгрупа – Belarusian" lang="be" hreflang="be" data-title="Падгрупа" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Подгрупа – Bulgarian" lang="bg" hreflang="bg" data-title="Подгрупа" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Subgrup" title="Subgrup – Catalan" lang="ca" hreflang="ca" data-title="Subgrup" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Podgrupa" title="Podgrupa – Czech" lang="cs" hreflang="cs" data-title="Podgrupa" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Undergruppe" title="Undergruppe – Danish" lang="da" hreflang="da" data-title="Undergruppe" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Untergruppe" title="Untergruppe – German" lang="de" hreflang="de" data-title="Untergruppe" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Subgrupo" title="Subgrupo – Spanish" lang="es" hreflang="es" data-title="Subgrupo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Subgrupo" title="Subgrupo – Esperanto" lang="eo" hreflang="eo" data-title="Subgrupo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%DB%8C%D8%B1%DA%AF%D8%B1%D9%88%D9%87" title="زیرگروه – Persian" lang="fa" hreflang="fa" data-title="زیرگروه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Sous-groupe" title="Sous-groupe – French" lang="fr" hreflang="fr" data-title="Sous-groupe" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Subgrupo" title="Subgrupo – Galician" lang="gl" hreflang="gl" data-title="Subgrupo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%80%EB%B6%84%EA%B5%B0" title="부분군 – Korean" lang="ko" hreflang="ko" data-title="부분군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Podgrupa" title="Podgrupa – Croatian" lang="hr" hreflang="hr" data-title="Podgrupa" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Subgrup" title="Subgrup – Indonesian" lang="id" hreflang="id" data-title="Subgrup" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Subgruppo" title="Subgruppo – Interlingua" lang="ia" hreflang="ia" data-title="Subgruppo" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sottogruppo" title="Sottogruppo – Italian" lang="it" hreflang="it" data-title="Sottogruppo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he badge-Q70894304 mw-list-item" title=""><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%AA-%D7%97%D7%91%D7%95%D7%A8%D7%94" title="תת-חבורה – Hebrew" lang="he" hreflang="he" data-title="תת-חבורה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/R%C3%A9szcsoport" title="Részcsoport – Hungarian" lang="hu" hreflang="hu" data-title="Részcsoport" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%89%E0%B4%AA%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D" title="ഉപഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ഉപഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ondergroep_(wiskunde)" title="Ondergroep (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Ondergroep (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%83%A8%E5%88%86%E7%BE%A4" title="部分群 – Japanese" lang="ja" hreflang="ja" data-title="部分群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Podgrupa" title="Podgrupa – Polish" lang="pl" hreflang="pl" data-title="Podgrupa" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Subgrupo" title="Subgrupo – Portuguese" lang="pt" hreflang="pt" data-title="Subgrupo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Subgrup" title="Subgrup – Romanian" lang="ro" hreflang="ro" data-title="Subgrup" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Подгруппа – Russian" lang="ru" hreflang="ru" data-title="Подгруппа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Subgroup" title="Subgroup – Simple English" lang="en-simple" hreflang="en-simple" data-title="Subgroup" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Podgrupa" title="Podgrupa – Slovak" lang="sk" hreflang="sk" data-title="Podgrupa" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Podgrupa" title="Podgrupa – Slovenian" lang="sl" hreflang="sl" data-title="Podgrupa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Подгрупа (математика) – Serbian" lang="sr" hreflang="sr" data-title="Подгрупа (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Podgrupa" title="Podgrupa – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Podgrupa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Aliryhm%C3%A4" title="Aliryhmä – Finnish" lang="fi" hreflang="fi" data-title="Aliryhmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Delgrupp" title="Delgrupp – Swedish" lang="sv" hreflang="sv" data-title="Delgrupp" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%89%E0%AE%9F%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="உட்குலம் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="உட்குலம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alt%C3%B6bek" title="Altöbek – Turkish" lang="tr" hreflang="tr" data-title="Altöbek" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%96%D0%B4%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Підгрупа – Ukrainian" lang="uk" hreflang="uk" data-title="Підгрупа" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_con" title="Nhóm con – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm con" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%AD%90%E7%BE%A3" title="子羣 – Cantonese" lang="yue" hreflang="yue" data-title="子羣" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh 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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;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><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></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><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 class="mw-selflink selflink">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></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> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></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"><i>p</i>-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> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</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, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>)</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(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></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> <p>In <a href="/wiki/Group_theory" title="Group theory">group theory</a>, a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. </p><p>Formally, given a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <span class="texhtml mvar" style="font-style:italic;">G</span> under a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> ∗, a <a href="/wiki/Subset" title="Subset">subset</a> <span class="texhtml mvar" style="font-style:italic;">H</span> of <span class="texhtml mvar" style="font-style:italic;">G</span> is called a <b>subgroup</b> of <span class="texhtml mvar" style="font-style:italic;">G</span> if <span class="texhtml mvar" style="font-style:italic;">H</span> also forms a group under the operation ∗. More precisely, <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span> if the <a href="/wiki/Restriction_(mathematics)" title="Restriction (mathematics)">restriction</a> of ∗ to <span class="texhtml"><i>H</i> × <i>H</i></span> is a group operation on <span class="texhtml mvar" style="font-style:italic;">H</span>. This is often denoted <span class="texhtml"><i>H</i> ≤ <i>G</i></span>, read as "<span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>". </p><p>The <b>trivial subgroup</b> of any group is the subgroup {<i>e</i>} consisting of just the identity element.<sup id="cite_ref-FOOTNOTEGallian201361_1-0" class="reference"><a href="#cite_note-FOOTNOTEGallian201361-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>A <b>proper subgroup</b> of a group <span class="texhtml mvar" style="font-style:italic;">G</span> is a subgroup <span class="texhtml mvar" style="font-style:italic;">H</span> which is a <a href="/wiki/Subset" title="Subset">proper subset</a> of <span class="texhtml mvar" style="font-style:italic;">G</span> (that is, <span class="texhtml"><i>H</i> ≠ <i>G</i></span>). This is often represented notationally by <span class="texhtml"><i>H</i> < <i>G</i></span>, read as "<span class="texhtml mvar" style="font-style:italic;">H</span> is a proper subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>". Some authors also exclude the trivial group from being proper (that is, <span class="texhtml"><i>H</i> ≠ {<i>e</i>}</span>).<sup id="cite_ref-FOOTNOTEHungerford197432_2-0" class="reference"><a href="#cite_note-FOOTNOTEHungerford197432-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEArtin201143_3-0" class="reference"><a href="#cite_note-FOOTNOTEArtin201143-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>If <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>, then <span class="texhtml mvar" style="font-style:italic;">G</span> is sometimes called an <b>overgroup</b> of <span class="texhtml mvar" style="font-style:italic;">H</span>. </p><p>The same definitions apply more generally when <span class="texhtml mvar" style="font-style:italic;">G</span> is an arbitrary <a href="/wiki/Semigroup" title="Semigroup">semigroup</a>, but this article will only deal with subgroups of groups. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Subgroup_tests">Subgroup tests</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=1" title="Edit section: Subgroup tests"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Suppose that <span class="texhtml mvar" style="font-style:italic;">G</span> is a group, and <span class="texhtml mvar" style="font-style:italic;">H</span> is a subset of <span class="texhtml mvar" style="font-style:italic;">G</span>. For now, assume that the group operation of <span class="texhtml mvar" style="font-style:italic;">G</span> is written multiplicatively, denoted by juxtaposition. </p> <ul><li>Then <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="texhtml mvar" style="font-style:italic;">H</span> is nonempty and <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under products and inverses. <i>Closed under products</i> means that for every <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> in <span class="texhtml mvar" style="font-style:italic;">H</span>, the product <span class="texhtml mvar" style="font-style:italic;">ab</span> is in <span class="texhtml mvar" style="font-style:italic;">H</span>. <i>Closed under inverses</i> means that for every <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">H</span>, the inverse <span class="texhtml"><i>a</i><sup>−1</sup></span> is in <span class="texhtml mvar" style="font-style:italic;">H</span>. These two conditions can be combined into one, that for every <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> in <span class="texhtml mvar" style="font-style:italic;">H</span>, the element <span class="texhtml"><i>ab</i><sup>−1</sup></span> is in <span class="texhtml mvar" style="font-style:italic;">H</span>, but it is more natural and usually just as easy to test the two closure conditions separately.<sup id="cite_ref-FOOTNOTEKurzweilStellmacher19984_4-0" class="reference"><a href="#cite_note-FOOTNOTEKurzweilStellmacher19984-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>When <span class="texhtml mvar" style="font-style:italic;">H</span> is <i>finite</i>, the test can be simplified: <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup if and only if it is nonempty and closed under products. These conditions alone imply that every element <span class="texhtml mvar" style="font-style:italic;">a</span> of <span class="texhtml mvar" style="font-style:italic;">H</span> generates a finite cyclic subgroup of <span class="texhtml mvar" style="font-style:italic;">H</span>, say of order <span class="texhtml mvar" style="font-style:italic;">n</span>, and then the inverse of <span class="texhtml mvar" style="font-style:italic;">a</span> is <span class="texhtml"><i>a</i><sup><i>n</i>−1</sup></span>.<sup id="cite_ref-FOOTNOTEKurzweilStellmacher19984_4-1" class="reference"><a href="#cite_note-FOOTNOTEKurzweilStellmacher19984-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul> <p>If the group operation is instead denoted by addition, then <i>closed under products</i> should be replaced by <i>closed under addition</i>, which is the condition that for every <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> in <span class="texhtml mvar" style="font-style:italic;">H</span>, the sum <span class="texhtml"><i>a</i> + <i>b</i></span> is in <span class="texhtml mvar" style="font-style:italic;">H</span>, and <i>closed under inverses</i> should be edited to say that for every <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">H</span>, the inverse <span class="texhtml">−<i>a</i></span> is in <span class="texhtml mvar" style="font-style:italic;">H</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_properties_of_subgroups">Basic properties of subgroups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=2" title="Edit section: Basic properties of subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Identity_element" title="Identity element">identity</a> of a subgroup is the identity of the group: if <span class="texhtml mvar" style="font-style:italic;">G</span> is a group with identity <span class="texhtml mvar" style="font-style:italic;">e<sub>G</sub></span>, and <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span> with identity <span class="texhtml mvar" style="font-style:italic;">e<sub>H</sub></span>, then <span class="texhtml"><i>e<sub>H</sub></i> = <i>e<sub>G</sub></i></span>.</li> <li>The <a href="/wiki/Inverse_element" title="Inverse element">inverse</a> of an element in a subgroup is the inverse of the element in the group: if <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of a group <span class="texhtml mvar" style="font-style:italic;">G</span>, and <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> are elements of <span class="texhtml mvar" style="font-style:italic;">H</span> such that <span class="texhtml"><i>ab</i> = <i>ba</i> = <i>e<sub>H</sub></i></span>, then <span class="texhtml"><i>ab</i> = <i>ba</i> = <i>e<sub>G</sub></i></span>.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">H</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>, then the inclusion map <span class="texhtml"><i>H</i> → <i>G</i></span> sending each element <span class="texhtml mvar" style="font-style:italic;">a</span> of <span class="texhtml mvar" style="font-style:italic;">H</span> to itself is a <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a>.</li> <li>The <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of subgroups <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> of <span class="texhtml mvar" style="font-style:italic;">G</span> is again a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>.<sup id="cite_ref-FOOTNOTEJacobson200941_5-0" class="reference"><a href="#cite_note-FOOTNOTEJacobson200941-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> For example, the intersection of the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis and <span class="texhtml mvar" style="font-style:italic;">y</span>-axis in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>⁠</span> under addition is the trivial subgroup. More generally, the intersection of an arbitrary collection of subgroups of <span class="texhtml mvar" style="font-style:italic;">G</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>.</li> <li>The <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of subgroups <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> is a subgroup if and only if <span class="texhtml"><i>A</i> ⊆ <i>B</i></span> or <span class="texhtml"><i>B</i> ⊆ <i>A</i></span>. A non-example: <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>∪</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b739fb93a00d45ce7b10fbd585b30d28baccd56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.008ex; height:2.176ex;" alt="{\displaystyle 2\mathbb {Z} \cup 3\mathbb {Z} }"></span>⁠</span> is not a subgroup of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}"></span>⁠</span> because 2 and 3 are elements of this subset whose sum, 5, is not in the subset. Similarly, the union of the <span class="texhtml mvar" style="font-style:italic;">x</span>-axis and the <span class="texhtml mvar" style="font-style:italic;">y</span>-axis in <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span>⁠</span> is not a subgroup of <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066b155c535a38739cc0c4b288324cbb7a4a227a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.379ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}.}"></span>⁠</span></li> <li>If <span class="texhtml mvar" style="font-style:italic;">S</span> is a subset of <span class="texhtml mvar" style="font-style:italic;">G</span>, then there exists a smallest subgroup containing <span class="texhtml mvar" style="font-style:italic;">S</span>, namely the intersection of all of subgroups containing <span class="texhtml mvar" style="font-style:italic;">S</span>; it is denoted by <span class="texhtml"><span class="nowrap">⟨<i>S</i>⟩</span></span> and is called the <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">subgroup generated by <span class="texhtml mvar" style="font-style:italic;">S</span></a>. An element of <span class="texhtml mvar" style="font-style:italic;">G</span> is in <span class="texhtml"><span class="nowrap">⟨<i>S</i>⟩</span></span> if and only if it is a finite product of elements of <span class="texhtml mvar" style="font-style:italic;">S</span> and their inverses, possibly repeated.<sup id="cite_ref-FOOTNOTEAsh2002_6-0" class="reference"><a href="#cite_note-FOOTNOTEAsh2002-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>Every element <span class="texhtml mvar" style="font-style:italic;">a</span> of a group <span class="texhtml mvar" style="font-style:italic;">G</span> generates a cyclic subgroup <span class="texhtml"><span class="nowrap">⟨<i>a</i>⟩</span></span>. If <span class="texhtml"><span class="nowrap">⟨<i>a</i>⟩</span></span> is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2120ebbc85f91df66c6de5446367bf9fd620844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"></span>⁠</span> (<a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">the integers <span class="texhtml">mod <i>n</i></span></a>) for some positive integer <span class="texhtml mvar" style="font-style:italic;">n</span>, then <span class="texhtml mvar" style="font-style:italic;">n</span> is the smallest positive integer for which <span class="texhtml"><i>a<sup>n</sup></i> = <i>e</i></span>, and <span class="texhtml mvar" style="font-style:italic;">n</span> is called the <i>order</i> of <span class="texhtml mvar" style="font-style:italic;">a</span>. If <span class="texhtml"><span class="nowrap">⟨<i>a</i>⟩</span></span> is isomorphic to <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}"></span>⁠</span> then <span class="texhtml mvar" style="font-style:italic;">a</span> is said to have <i>infinite order</i>.</li> <li>The subgroups of any given group form a <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a> under inclusion, called the <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a>. (While the <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> here is the usual set-theoretic intersection, the <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a> of a set of subgroups is the subgroup <i>generated by</i> the set-theoretic union of the subgroups, not the set-theoretic union itself.) If <span class="texhtml mvar" style="font-style:italic;">e</span> is the identity of <span class="texhtml mvar" style="font-style:italic;">G</span>, then the trivial group <span class="texhtml">{<i>e</i>} </span> is the <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">minimum</a> subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span>, while the <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">maximum</a> subgroup is the group <span class="texhtml mvar" style="font-style:italic;">G</span> itself.</li></ul> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Left_cosets_of_Z_2_in_Z_8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Left_cosets_of_Z_2_in_Z_8.svg/220px-Left_cosets_of_Z_2_in_Z_8.svg.png" decoding="async" width="220" height="252" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Left_cosets_of_Z_2_in_Z_8.svg/330px-Left_cosets_of_Z_2_in_Z_8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Left_cosets_of_Z_2_in_Z_8.svg/440px-Left_cosets_of_Z_2_in_Z_8.svg.png 2x" data-file-width="238" data-file-height="273" /></a><figcaption><span class="texhtml mvar" style="font-style:italic;">G</span> is the group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /8\mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /8\mathbb {Z} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f06e0292a3b67b96b7727c3a5b1c4c6990afecfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /8\mathbb {Z} ,}"></span> the <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">integers mod 8</a> under addition. The subgroup <span class="texhtml mvar" style="font-style:italic;">H</span> contains only 0 and 4, and is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/876ee1031b5a964b0cd5ba8c4109647f28e3522a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} .}"></span> There are four left cosets of <span class="texhtml mvar" style="font-style:italic;">H</span>: <span class="texhtml mvar" style="font-style:italic;">H</span> itself, <span class="texhtml">1 + <i>H</i></span>, <span class="texhtml">2 + <i>H</i></span>, and <span class="texhtml">3 + <i>H</i></span> (written using additive notation since this is an <a href="/wiki/Abelian_group" title="Abelian group">additive group</a>). Together they partition the entire group <span class="texhtml mvar" style="font-style:italic;">G</span> into equal-size, non-overlapping sets. The index <span class="texhtml">[<i>G</i> : <i>H</i>]</span> is 4.</figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Cosets_and_Lagrange's_theorem"><span id="Cosets_and_Lagrange.27s_theorem"></span>Cosets and Lagrange's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=3" title="Edit section: Cosets and Lagrange's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Coset" title="Coset">Coset</a> and <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem (group theory)</a></div> <p>Given a subgroup <span class="texhtml mvar" style="font-style:italic;">H</span> and some <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">G</span>, we define the <b>left <a href="/wiki/Coset" title="Coset">coset</a></b> <span class="texhtml"><i>aH</i> = {<i>ah</i> : <i>h</i> in <i>H</i>}.</span> Because <span class="texhtml mvar" style="font-style:italic;">a</span> is invertible, the map <span class="texhtml">φ : <i>H</i> → <i>aH</i></span> given by <span class="texhtml">φ(<i>h</i>) = <i>ah</i></span> is a <a href="/wiki/Bijection" title="Bijection">bijection</a>. Furthermore, every element of <span class="texhtml mvar" style="font-style:italic;">G</span> is contained in precisely one left coset of <span class="texhtml mvar" style="font-style:italic;">H</span>; the left cosets are the equivalence classes corresponding to the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> <span class="texhtml"><i>a</i><sub>1</sub> ~ <i>a</i><sub>2</sub></span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}^{-1}a_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}^{-1}a_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e19f839ff7ba65552eb1d6e32f69240413fac0a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.847ex; height:3.343ex;" alt="{\displaystyle a_{1}^{-1}a_{2}}"></span>⁠</span> is in <span class="texhtml mvar" style="font-style:italic;">H</span>. The number of left cosets of <span class="texhtml mvar" style="font-style:italic;">H</span> is called the <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> of <span class="texhtml mvar" style="font-style:italic;">H</span> in <span class="texhtml mvar" style="font-style:italic;">G</span> and is denoted by <span class="texhtml">[<i>G</i> : <i>H</i>]</span>. </p><p><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a> states that for a finite group <span class="texhtml mvar" style="font-style:italic;">G</span> and a subgroup <span class="texhtml mvar" style="font-style:italic;">H</span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [G:H]={|G| \over |H|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]={|G| \over |H|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e64b76d0605e6c7891273e65da28b5e431d4ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.413ex; height:6.509ex;" alt="{\displaystyle [G:H]={|G| \over |H|}}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">G</span>|</span> and <span class="texhtml mvar" style="font-style:italic;">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">H</span>|</span> denote the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">orders</a> of <span class="texhtml mvar" style="font-style:italic;">G</span> and <span class="texhtml mvar" style="font-style:italic;">H</span>, respectively. In particular, the order of every subgroup of <span class="texhtml mvar" style="font-style:italic;">G</span> (and the order of every element of <span class="texhtml mvar" style="font-style:italic;">G</span>) must be a <a href="/wiki/Divisor" title="Divisor">divisor</a> of <span class="texhtml mvar" style="font-style:italic;">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;">G</span>|</span>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEDummitFoote200490_8-0" class="reference"><a href="#cite_note-FOOTNOTEDummitFoote200490-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p><b>Right cosets</b> are defined analogously: <span class="texhtml"><i>Ha</i> = {<i>ha</i> : <i>h</i> in <i>H</i>}.</span> They are also the equivalence classes for a suitable equivalence relation and their number is equal to <span class="texhtml">[<i>G</i> : <i>H</i>]</span>. </p><p>If <span class="texhtml"><i>aH</i> = <i>Ha</i></span> for every <span class="texhtml mvar" style="font-style:italic;">a</span> in <span class="texhtml mvar" style="font-style:italic;">G</span>, then <span class="texhtml mvar" style="font-style:italic;">H</span> is said to be a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>. Every subgroup of index 2 is normal: the left cosets, and also the right cosets, are simply the subgroup and its complement. More generally, if <span class="texhtml mvar" style="font-style:italic;">p</span> is the lowest prime dividing the order of a finite group <span class="texhtml mvar" style="font-style:italic;">G</span>, then any subgroup of index <span class="texhtml mvar" style="font-style:italic;">p</span> (if such exists) is normal. </p> <div class="mw-heading mw-heading2"><h2 id="Example:_Subgroups_of_Z8">Example: Subgroups of Z<sub>8</sub></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=4" title="Edit section: Example: Subgroups of Z8"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">G</span> be the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> <span class="texhtml">Z<sub>8</sub></span> whose elements are </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mn>0</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>7</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faa8a8d72d9c053af0df038cf9bf71c26df65f41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.787ex; height:2.843ex;" alt="{\displaystyle G=\left\{0,4,2,6,1,5,3,7\right\}}"></span></dd></dl> <p>and whose group operation is <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">addition modulo 8</a>. Its <a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a> is </p> <table class="wikitable" style="color:blue;"> <tbody><tr> <th style="background-color:#FFFFAA; color:black;">+ </th> <th style="background-color:#FFFFAA; color:orange;">0 </th> <th style="background-color:#FFFFAA; color:orange;">4 </th> <th style="background-color:#FFFFAA; color:red;">2 </th> <th style="background-color:#FFFFAA; color:red;">6 </th> <th style="background-color:#FFFFAA;">1 </th> <th style="background-color:#FFFFAA;">5 </th> <th style="background-color:#FFFFAA;">3 </th> <th style="background-color:#FFFFAA;">7 </th></tr> <tr> <th style="background:#FFFFAA; color:orange;">0 </th> <td style="color:orange;">0</td> <td style="color:orange;">4 </td> <td style="color:red;">2</td> <td style="color:red;">6 </td> <td>1</td> <td>5</td> <td>3</td> <td>7 </td></tr> <tr> <th style="background:#FFFFAA; color:orange;">4 </th> <td style="color:orange;">4</td> <td style="color:orange;">0 </td> <td style="color:red;">6</td> <td style="color:red;">2 </td> <td>5</td> <td>1</td> <td>7</td> <td>3 </td></tr> <tr> <th style="background:#FFFFAA; color:red;">2 </th> <td style="color:red;">2</td> <td style="color:red;">6</td> <td style="color:red;">4</td> <td style="color:red;">0 </td> <td>3</td> <td>7</td> <td>5</td> <td>1 </td></tr> <tr> <th style="background:#FFFFAA; color:red;">6 </th> <td style="color:red;">6</td> <td style="color:red;">2</td> <td style="color:red;">0</td> <td style="color:red;">4 </td> <td>7</td> <td>3</td> <td>1</td> <td>5 </td></tr> <tr> <th style="background-color:#FFFFAA;">1 </th> <td>1</td> <td>5</td> <td>3</td> <td>7</td> <td>2</td> <td>6</td> <td>4</td> <td>0 </td></tr> <tr> <th style="background-color:#FFFFAA;">5 </th> <td>5</td> <td>1</td> <td>7</td> <td>3</td> <td>6</td> <td>2</td> <td>0</td> <td>4 </td></tr> <tr> <th style="background-color:#FFFFAA;">3 </th> <td>3</td> <td>7</td> <td>5</td> <td>1</td> <td>4</td> <td>0</td> <td>6</td> <td>2 </td></tr> <tr> <th style="background-color:#FFFFAA;">7 </th> <td>7</td> <td>3</td> <td>1</td> <td>5</td> <td>0</td> <td>4</td> <td>2</td> <td>6 </td></tr></tbody></table> <p>This group has two nontrivial subgroups: <span class="texhtml"><span style="color:orange; font-size:100%; line-height:1;" title="Orange">■</span> <i>J</i> = {0, 4} </span> and <span class="texhtml"><span style="color:red; font-size:100%; line-height:1;" title="Red">■</span> <i>H</i> = {0, 4, 2, 6} </span>, where <span class="texhtml mvar" style="font-style:italic;">J</span> is also a subgroup of <span class="texhtml mvar" style="font-style:italic;">H</span>. The Cayley table for <span class="texhtml mvar" style="font-style:italic;">H</span> is the top-left quadrant of the Cayley table for <span class="texhtml mvar" style="font-style:italic;">G</span>; The Cayley table for <span class="texhtml mvar" style="font-style:italic;">J</span> is the top-left quadrant of the Cayley table for <span class="texhtml mvar" style="font-style:italic;">H</span>. The group <span class="texhtml mvar" style="font-style:italic;">G</span> is <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a>, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.<sup id="cite_ref-FOOTNOTEGallian201381_9-0" class="reference"><a href="#cite_note-FOOTNOTEGallian201381-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Example:_Subgroups_of_S4">Example: Subgroups of S<sub>4</sub><span class="anchor" id="Subgroups_of_S4"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=5" title="Edit section: Example: Subgroups of S4"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="texhtml">S<sub>4</sub></span> is the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> whose elements correspond to the <a href="/wiki/Permutation" title="Permutation">permutations</a> of 4 elements.<br /> Below are all its subgroups, ordered by cardinality.<br /> Each group <small>(except those of cardinality 1 and 2)</small> is represented by its <a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a>. </p> <div class="mw-heading mw-heading3"><h3 id="24_elements">24 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=6" title="Edit section: 24 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Like each group, <span class="texhtml">S<sub>4</sub></span> is a subgroup of itself. </p> <table style="width:100%"> <tbody><tr> <td style="vertical-align:top;"><figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_4;_Cayley_table;_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Symmetric_group_4%3B_Cayley_table%3B_numbers.svg/595px-Symmetric_group_4%3B_Cayley_table%3B_numbers.svg.png" decoding="async" width="595" height="595" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Symmetric_group_4%3B_Cayley_table%3B_numbers.svg/893px-Symmetric_group_4%3B_Cayley_table%3B_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Symmetric_group_4%3B_Cayley_table%3B_numbers.svg/1190px-Symmetric_group_4%3B_Cayley_table%3B_numbers.svg.png 2x" data-file-width="744" data-file-height="744" /></a><figcaption>Symmetric group <span class="texhtml">S<sub>4</sub></span></figcaption></figure> </td> <td style="vertical-align:top;"> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:443px;max-width:443px"><div class="trow"><div class="tsingle" style="width:252px;max-width:252px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Symmetric_group_S4;_lattice_of_subgroups_Hasse_diagram;_all_30_subgroups.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_all_30_subgroups.svg/250px-Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_all_30_subgroups.svg.png" decoding="async" width="250" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_all_30_subgroups.svg/375px-Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_all_30_subgroups.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_all_30_subgroups.svg/500px-Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_all_30_subgroups.svg.png 2x" data-file-width="1054" data-file-height="1012" /></a></span></div><div class="thumbcaption">All 30 subgroups</div></div><div class="tsingle" style="width:187px;max-width:187px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Symmetric_group_S4;_lattice_of_subgroups_Hasse_diagram;_11_different_cycle_graphs.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_11_different_cycle_graphs.svg/185px-Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_11_different_cycle_graphs.svg.png" decoding="async" width="185" height="240" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_11_different_cycle_graphs.svg/278px-Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_11_different_cycle_graphs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_11_different_cycle_graphs.svg/370px-Symmetric_group_S4%3B_lattice_of_subgroups_Hasse_diagram%3B_11_different_cycle_graphs.svg.png 2x" data-file-width="840" data-file-height="1091" /></a></span></div><div class="thumbcaption">Simplified</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagrams</a> of the <a href="/wiki/Lattice_of_subgroups" title="Lattice of subgroups">lattice of subgroups</a> of <span class="texhtml">S<sub>4</sub></span></div></div></div></div> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="12_elements">12 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=7" title="Edit section: 12 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> contains only the <a href="/wiki/Parity_of_a_permutation" title="Parity of a permutation">even permutations</a>.<br /> It is one of the two nontrivial proper <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroups</a> of <span class="texhtml">S<sub>4</sub></span>. <small>(The other one is its Klein subgroup.)</small> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Alternating_group_4;_Cayley_table;_numbers.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Alternating_group_4%3B_Cayley_table%3B_numbers.svg/323px-Alternating_group_4%3B_Cayley_table%3B_numbers.svg.png" decoding="async" width="323" height="323" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Alternating_group_4%3B_Cayley_table%3B_numbers.svg/485px-Alternating_group_4%3B_Cayley_table%3B_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8b/Alternating_group_4%3B_Cayley_table%3B_numbers.svg/646px-Alternating_group_4%3B_Cayley_table%3B_numbers.svg.png 2x" data-file-width="404" data-file-height="404" /></a><figcaption>Alternating group <span class="texhtml">A<sub>4</sub></span><br /><br />Subgroups:<br /><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><br /><span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,3,4).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span><span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,11,19).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span> <span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,15,20).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span> <span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,8,12).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span></figcaption></figure> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="8_elements">8 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=8" title="Edit section: 8 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Dihedral_group_of_order_8;_Cayley_table_(element_orders_1,2,2,2,2,4,4,2);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C2%2C2%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/233px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C2%2C2%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png" decoding="async" width="233" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C2%2C2%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/350px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C2%2C2%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C2%2C2%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/466px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C2%2C2%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="291" data-file-height="291" /></a><figcaption><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> <a href="/wiki/Dihedral_group_of_order_8" class="mw-redirect" title="Dihedral group of order 8">of order 8</a><br /><br />Subgroups:<br /><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,1,6,7).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><span typeof="mw:File"><a href="/wiki/File:Cyclic_group_4;_Cayley_table_(element_orders_1,2,4,4);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg/70px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg/105px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg/140px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span></figcaption></figure></td> <td> </td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Dihedral_group_of_order_8;_Cayley_table_(element_orders_1,2,2,4,2,2,4,2);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C2%2C2%2C4%2C2%29%3B_subgroup_of_S4.svg/233px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C2%2C2%2C4%2C2%29%3B_subgroup_of_S4.svg.png" decoding="async" width="233" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C2%2C2%2C4%2C2%29%3B_subgroup_of_S4.svg/350px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C2%2C2%2C4%2C2%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C2%2C2%2C4%2C2%29%3B_subgroup_of_S4.svg/466px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C2%2C2%2C4%2C2%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="291" data-file-height="291" /></a><figcaption>Dihedral group of order 8<br /><br />Subgroups:<br /><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,5,14,16).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><span typeof="mw:File"><a href="/wiki/File:Cyclic_group_4;_Cayley_table_(element_orders_1,4,2,4);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg/70px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg/105px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg/140px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span></figcaption></figure></td> <td> </td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Dihedral_group_of_order_8;_Cayley_table_(element_orders_1,2,2,4,4,2,2,2);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C4%2C2%2C2%2C2%29%3B_subgroup_of_S4.svg/233px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C4%2C2%2C2%2C2%29%3B_subgroup_of_S4.svg.png" decoding="async" width="233" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C4%2C2%2C2%2C2%29%3B_subgroup_of_S4.svg/350px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C4%2C2%2C2%2C2%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4e/Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C4%2C2%2C2%2C2%29%3B_subgroup_of_S4.svg/466px-Dihedral_group_of_order_8%3B_Cayley_table_%28element_orders_1%2C2%2C2%2C4%2C4%2C2%2C2%2C2%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="291" data-file-height="291" /></a><figcaption>Dihedral group of order 8<br /><br />Subgroups:<br /><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,2,21,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><span typeof="mw:File"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/70px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/105px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/140px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span><span typeof="mw:File"><a href="/wiki/File:Cyclic_group_4;_Cayley_table_(element_orders_1,4,4,2);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/70px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png" decoding="async" width="70" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/105px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/140px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="177" data-file-height="177" /></a></span></figcaption></figure> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="6_elements">6 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=9" title="Edit section: 6 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,1,2,3,4,5).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C2%2C3%2C4%2C5%29.svg/187px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C2%2C3%2C4%2C5%29.svg.png" decoding="async" width="187" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C2%2C3%2C4%2C5%29.svg/281px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C2%2C3%2C4%2C5%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C2%2C3%2C4%2C5%29.svg/374px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C2%2C3%2C4%2C5%29.svg.png 2x" data-file-width="234" data-file-height="234" /></a><figcaption><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> <span class="texhtml"><a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">S<sub>3</sub></a></span><br /><br />Subgroup:<span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,3,4).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,5,6,11,19,21).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C6%2C11%2C19%2C21%29.svg/187px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C6%2C11%2C19%2C21%29.svg.png" decoding="async" width="187" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C6%2C11%2C19%2C21%29.svg/281px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C6%2C11%2C19%2C21%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C6%2C11%2C19%2C21%29.svg/374px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C6%2C11%2C19%2C21%29.svg.png 2x" data-file-width="234" data-file-height="234" /></a><figcaption>Symmetric group <span class="texhtml">S<sub>3</sub></span><br /><br />Subgroup:<span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,11,19).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,1,14,15,20,21).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15%2C20%2C21%29.svg/187px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15%2C20%2C21%29.svg.png" decoding="async" width="187" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15%2C20%2C21%29.svg/281px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15%2C20%2C21%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15%2C20%2C21%29.svg/374px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C14%2C15%2C20%2C21%29.svg.png 2x" data-file-width="234" data-file-height="234" /></a><figcaption>Symmetric group <span class="texhtml">S<sub>3</sub></span><br /><br />Subgroup:<span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,15,20).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,2,6,8,12,14).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C6%2C8%2C12%2C14%29.svg/187px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C6%2C8%2C12%2C14%29.svg.png" decoding="async" width="187" height="187" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C6%2C8%2C12%2C14%29.svg/281px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C6%2C8%2C12%2C14%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C6%2C8%2C12%2C14%29.svg/374px-Symmetric_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C6%2C8%2C12%2C14%29.svg.png 2x" data-file-width="234" data-file-height="234" /></a><figcaption>Symmetric group <span class="texhtml">S<sub>3</sub></span><br /><br />Subgroup:<span typeof="mw:File"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,8,12).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/60px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/90px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a></span></figcaption></figure> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="4_elements">4 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=10" title="Edit section: 4 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table> <tbody><tr style="vertical-align: top;"> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,1,6,7).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/142px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/213px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/284px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption><a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,5,14,16).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg/142px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg/213px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg/284px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C5%2C14%2C16%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Klein four-group</figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,2,21,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg/142px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg/213px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg/284px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C2%2C21%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Klein four-group</figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/142px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/213px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/284px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Klein four-group<br /><small>(<a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>)</small></figcaption></figure> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_4;_Cayley_table_(element_orders_1,2,4,4);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg/142px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg/213px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg/284px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C2%2C4%2C4%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> <span class="texhtml">Z<sub>4</sub></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_4;_Cayley_table_(element_orders_1,4,2,4);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg/142px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg/213px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg/284px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C2%2C4%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Cyclic group <span class="texhtml">Z<sub>4</sub></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_4;_Cayley_table_(element_orders_1,4,4,2);_subgroup_of_S4.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/142px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/40/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/213px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/40/Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg/284px-Cyclic_group_4%3B_Cayley_table_%28element_orders_1%2C4%2C4%2C2%29%3B_subgroup_of_S4.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Cyclic group <span class="texhtml">Z<sub>4</sub></span></figcaption></figure> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="3_elements">3 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=11" title="Edit section: 3 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table> <tbody><tr> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,3,4).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/180px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg/240px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C3%2C4%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a><figcaption><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> <span class="texhtml">Z<sub>3</sub></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,11,19).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/180px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg/240px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C11%2C19%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a><figcaption>Cyclic group <span class="texhtml">Z<sub>3</sub></span></figcaption></figure></td> <td><figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,15,20).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/180px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg/240px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C15%2C20%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a><figcaption>Cyclic group <span class="texhtml">Z<sub>3</sub></span></figcaption></figure></td> <td> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cyclic_group_3;_Cayley_table;_subgroup_of_S4_(elements_0,8,12).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/120px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png" decoding="async" width="120" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/180px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg/240px-Cyclic_group_3%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C8%2C12%29.svg.png 2x" data-file-width="149" data-file-height="149" /></a><figcaption>Cyclic group <span class="texhtml">Z<sub>3</sub></span></figcaption></figure> </td></tr></tbody></table> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="2_elements">2 elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=12" title="Edit section: 2 elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each permutation <span class="texhtml mvar" style="font-style:italic;">p</span> of order 2 generates a subgroup <span class="texhtml">{1, <i>p</i></span>}. These are the permutations that have only 2-cycles:<br /> </p> <ul><li>There are the 6 <a href="/wiki/Cyclic_permutation#Transpositions" title="Cyclic permutation">transpositions</a> with one 2-cycle.   <small>(green background)</small></li> <li>And 3 permutations with two 2-cycles.   <small>(white background, bold numbers)</small></li></ul> <div class="mw-heading mw-heading3"><h3 id="1_element">1 element</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=13" title="Edit section: 1 element"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Trivial_group" title="Trivial group">trivial subgroup</a> is the unique subgroup of order 1. </p> <div class="mw-heading mw-heading2"><h2 id="Other_examples">Other examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=14" title="Edit section: Other examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The even integers form a subgroup <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c112372c16743e8c02db5eac5c6de0659841bec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle 2\mathbb {Z} }"></span>⁠</span> of the <a href="/wiki/Integer_ring" class="mw-redirect" title="Integer ring">integer ring</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} :}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/538290647c2ff78ecdf859af856615d94a74e137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.842ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} :}"></span>⁠</span> the sum of two even integers is even, and the negative of an even integer is even.</li> <li>An <a href="/wiki/Ideal_(ring_theory)#Definitions" title="Ideal (ring theory)">ideal</a> in a ring <span class="texhtml mvar" style="font-style:italic;">R</span> is a subgroup of the additive group of <span class="texhtml mvar" style="font-style:italic;">R</span>.</li> <li>A <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> is a subgroup of the additive group of vectors.</li> <li>In an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>, the elements of finite <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> form a subgroup called the <a href="/wiki/Torsion_subgroup" title="Torsion subgroup">torsion subgroup</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Cartan_subgroup" title="Cartan subgroup">Cartan subgroup</a></li> <li><a href="/wiki/Fitting_subgroup" title="Fitting subgroup">Fitting subgroup</a></li> <li><a href="/wiki/Fixed-point_subgroup" title="Fixed-point subgroup">Fixed-point subgroup</a></li> <li><a href="/wiki/Fully_normalized_subgroup" title="Fully normalized subgroup">Fully normalized subgroup</a></li> <li><a href="/wiki/Stable_subgroup" class="mw-redirect" title="Stable subgroup">Stable subgroup</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-FOOTNOTEGallian201361-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGallian201361_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGallian2013">Gallian 2013</a>, p. 61.</span> </li> <li id="cite_note-FOOTNOTEHungerford197432-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHungerford197432_2-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHungerford1974">Hungerford 1974</a>, p. 32.</span> </li> <li id="cite_note-FOOTNOTEArtin201143-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEArtin201143_3-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFArtin2011">Artin 2011</a>, p. 43.</span> </li> <li id="cite_note-FOOTNOTEKurzweilStellmacher19984-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-FOOTNOTEKurzweilStellmacher19984_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTEKurzweilStellmacher19984_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFKurzweilStellmacher1998">Kurzweil & Stellmacher 1998</a>, p. 4.</span> </li> <li id="cite_note-FOOTNOTEJacobson200941-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEJacobson200941_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson 2009</a>, p. 41.</span> </li> <li id="cite_note-FOOTNOTEAsh2002-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEAsh2002_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFAsh2002">Ash 2002</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">See a <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=TCcSZEL_3CQ">didactic proof in this video</a>.</span> </li> <li id="cite_note-FOOTNOTEDummitFoote200490-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEDummitFoote200490_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, p. 90.</span> </li> <li id="cite_note-FOOTNOTEGallian201381-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEGallian201381_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFGallian2013">Gallian 2013</a>, p. 81.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgroup&action=edit&section=17" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJacobson2009" class="citation cs2"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009), <i>Basic algebra</i>, vol. 1 (2nd ed.), Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47189-1" title="Special:BookSources/978-0-486-47189-1"><bdi>978-0-486-47189-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.edition=2nd&rft.pub=Dover&rft.date=2009&rft.isbn=978-0-486-47189-1&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHungerford1974" class="citation cs2"><a href="/wiki/Thomas_W._Hungerford" title="Thomas W. Hungerford">Hungerford, Thomas</a> (1974), <i>Algebra</i> (1st ed.), Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387905181" title="Special:BookSources/9780387905181"><bdi>9780387905181</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=1st&rft.pub=Springer-Verlag&rft.date=1974&rft.isbn=9780387905181&rft.aulast=Hungerford&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin2011" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (2011), <i>Algebra</i> (2nd ed.), Prentice Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780132413770" title="Special:BookSources/9780132413770"><bdi>9780132413770</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=2nd&rft.pub=Prentice+Hall&rft.date=2011&rft.isbn=9780132413770&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDummitFoote2004" class="citation book cs1">Dummit, David S.; Foote, Richard M. (2004). <i>Abstract algebra</i> (3rd ed.). Hoboken, NJ: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471452348" title="Special:BookSources/9780471452348"><bdi>9780471452348</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/248917264">248917264</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+algebra&rft.place=Hoboken%2C+NJ&rft.edition=3rd&rft.pub=Wiley&rft.date=2004&rft_id=info%3Aoclcnum%2F248917264&rft.isbn=9780471452348&rft.aulast=Dummit&rft.aufirst=David+S.&rft.au=Foote%2C+Richard+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallian2013" class="citation book cs1"><a href="/wiki/Joseph_Gallian" title="Joseph Gallian">Gallian, Joseph A.</a> (2013). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/807255720"><i>Contemporary abstract algebra</i></a> (8th ed.). Boston, MA: Brooks/Cole Cengage Learning. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-133-59970-8" title="Special:BookSources/978-1-133-59970-8"><bdi>978-1-133-59970-8</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/807255720">807255720</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+abstract+algebra&rft.place=Boston%2C+MA&rft.edition=8th&rft.pub=Brooks%2FCole+Cengage+Learning&rft.date=2013&rft_id=info%3Aoclcnum%2F807255720&rft.isbn=978-1-133-59970-8&rft.aulast=Gallian&rft.aufirst=Joseph+A.&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F807255720&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKurzweilStellmacher1998" class="citation book cs1">Kurzweil, Hans; Stellmacher, Bernd (1998). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/978-3-642-58816-7"><i>Theorie der endlichen Gruppen</i></a>. Springer-Lehrbuch. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-58816-7">10.1007/978-3-642-58816-7</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theorie+der+endlichen+Gruppen&rft.series=Springer-Lehrbuch&rft.date=1998&rft_id=info%3Adoi%2F10.1007%2F978-3-642-58816-7&rft.aulast=Kurzweil&rft.aufirst=Hans&rft.au=Stellmacher%2C+Bernd&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2F978-3-642-58816-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAsh2002" class="citation book cs1">Ash, Robert B. (2002). <a rel="nofollow" class="external text" href="https://faculty.math.illinois.edu/~r-ash/Algebra.html"><i>Abstract Algebra: The Basic Graduate Year</i></a>. Department of Mathematics University of Illinois.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra%3A+The+Basic+Graduate+Year&rft.pub=Department+of+Mathematics+University+of+Illinois&rft.date=2002&rft.aulast=Ash&rft.aufirst=Robert+B.&rft_id=https%3A%2F%2Ffaculty.math.illinois.edu%2F~r-ash%2FAlgebra.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASubgroup" class="Z3988"></span></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 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.navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Groups" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="background: #ceecee"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_navbox" title="Template:Group navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_navbox" title="Template talk:Group navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_navbox" title="Special:EditPage/Template:Group navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Groups" style="font-size:114%;margin:0 4em"><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Groups</a></div></th></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Group_theory" title="Group theory">Basic notions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Commutator_subgroup" title="Commutator subgroup">Commutator subgroup</a></li> <li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphism</a></li> <li>(<a href="/wiki/Semidirect_product" title="Semidirect product">Semi-</a>) <a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">direct sum</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%">Types of groups</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">Abelian groups</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic groups</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">Infinite group</a></li> <li><a href="/wiki/Simple_group" title="Simple group">Simple groups</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">Solvable groups</a></li> <li><a href="/wiki/Symmetry_group" title="Symmetry group">Symmetry group</a></li> <li><a href="/wiki/Space_group" title="Space group">Space group</a></li> <li><a href="/wiki/Point_group" title="Point group">Point group</a></li> <li><a href="/wiki/Wallpaper_group" title="Wallpaper group">Wallpaper group</a></li> <li><a href="/wiki/Trivial_group" title="Trivial group">Trivial group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <dl><dt><a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></dt> <dd><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub>n</sub></dd> <dd><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub>n</sub></dd></dl> <dl><dt><a href="/wiki/Sporadic_group" title="Sporadic group">Sporadic groups</a></dt> <dd><a href="/wiki/Mathieu_group" title="Mathieu group">Mathieu group</a> M<sub>11..12</sub>,M<sub>22..24</sub></dd> <dd><a href="/wiki/Conway_group" title="Conway group">Conway group</a> Co<sub>1..3</sub></dd> <dd>Janko groups <a href="/wiki/Janko_group_J1" title="Janko group J1">J<sub>1</sub></a>, <a href="/wiki/Janko_group_J2" title="Janko group J2">J<sub>2</sub></a>, <a href="/wiki/Janko_group_J3" title="Janko group J3">J<sub>3</sub></a>, <a href="/wiki/Janko_group_J4" title="Janko group J4">J<sub>4</sub></a></dd> <dd><a href="/wiki/Fischer_group" title="Fischer group">Fischer group</a> F<sub>22..24</sub></dd> <dd><a href="/wiki/Baby_monster_group" title="Baby monster group">Baby monster group</a> B</dd> <dd><a href="/wiki/Monster_group" title="Monster group">Monster group</a> M</dd></dl> <dl><dt>Other finite groups</dt> <dd><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> <i>S</i><sub><i>n</i></sub></dd> <dd><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> <i>D</i><sub><i>n</i></sub></dd> <dd><a href="/wiki/Rubik%27s_Cube_group" title="Rubik's Cube group">Rubik's Cube group</a></dd></dl> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%"><a href="/wiki/Lie_group" title="Lie group">Lie groups</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear group</a> GL(n)</li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear group</a> SL(n)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal group</a> O(n)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Special orthogonal group</a> SO(n)</li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary group</a> U(n)</li> <li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary group</a> SU(n)</li> <li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic group</a> Sp(n)</li></ul> <dl><dt><a href="/wiki/Exceptional_Lie_groups" class="mw-redirect" title="Exceptional Lie groups">Exceptional Lie groups</a></dt> <dd><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></dd> <dd><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></dd> <dd><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></dd> <dd><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></dd> <dd><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></dd></dl> <ul><li><a href="/wiki/Circle_group" title="Circle group">Circle group</a></li> <li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="background: #ceecee;width:1%">Infinite dimensional groups</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism group</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop group</a></li> <li><a href="/wiki/Quantum_group" title="Quantum group">Quantum group</a></li> <li>O(∞)</li> <li>SU(∞)</li> <li>Sp(∞)</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2" style="background: #ceecee;font-weight:bold;"><div> <ul><li><a href="/wiki/History_of_group_theory" title="History of group theory">History</a></li> <li><a href="/wiki/Group_theory#Applications_of_group_theory" title="Group theory">Applications</a></li> <li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: 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