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<span>Toggle Existence of subgroups of given order subsection</span> </button> <ul id="toc-Existence_of_subgroups_of_given_order-sublist" class="vector-toc-list"> <li id="toc-Counterexample_of_the_converse_of_Lagrange's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Counterexample_of_the_converse_of_Lagrange's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Counterexample of the converse of Lagrange's theorem</span> </div> </a> <ul id="toc-Counterexample_of_the_converse_of_Lagrange's_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-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">6</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">7</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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" 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Available in 31 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-31" 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">31 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/%D9%85%D8%A8%D8%B1%D9%87%D9%86%D8%A9_%D9%84%D8%A7%D8%BA%D8%B1%D8%A7%D9%86%D8%AC_(%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%B2%D9%85%D8%B1)" title="مبرهنة لاغرانج (نظرية الزمر) – Arabic" lang="ar" hreflang="ar" data-title="مبرهنة لاغرانج (نظرية الزمر)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teorema_de_Lagrange_(%C3%A0lgebra)" title="Teorema de Lagrange (àlgebra) – Catalan" lang="ca" hreflang="ca" data-title="Teorema de Lagrange (àlgebra)" 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/Lagrangeova_v%C4%9Bta_(teorie_grup)" title="Lagrangeova věta (teorie grup) – Czech" lang="cs" hreflang="cs" data-title="Lagrangeova věta (teorie grup)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Satz_von_Lagrange" title="Satz von Lagrange – German" lang="de" hreflang="de" data-title="Satz von Lagrange" 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/Teorema_de_Lagrange_(teor%C3%ADa_de_grupos)" title="Teorema de Lagrange (teoría de grupos) – Spanish" lang="es" hreflang="es" data-title="Teorema de Lagrange (teoría de grupos)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%82%D8%B6%DB%8C%D9%87_%D9%84%D8%A7%DA%AF%D8%B1%D8%A7%D9%86%DA%98_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D9%88%D9%87%E2%80%8C%D9%87%D8%A7)" title="قضیه لاگرانژ (نظریه گروه‌ها) – Persian" lang="fa" hreflang="fa" data-title="قضیه لاگرانژ (نظریه گروه‌ها)" 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/Th%C3%A9or%C3%A8me_de_Lagrange_sur_les_groupes" title="Théorème de Lagrange sur les groupes – French" lang="fr" hreflang="fr" data-title="Théorème de Lagrange sur les groupes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%9D%BC%EA%B7%B8%EB%9E%91%EC%A3%BC_%EC%A0%95%EB%A6%AC_(%EA%B5%B0%EB%A1%A0)" title="라그랑주 정리 (군론) – Korean" lang="ko" hreflang="ko" data-title="라그랑주 정리 (군론)" 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/Lagrangeov_teorem_(teorija_grupa)" title="Lagrangeov teorem (teorija grupa) – Croatian" lang="hr" hreflang="hr" data-title="Lagrangeov teorem (teorija grupa)" 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/Teorema_Lagrange_(teori_grup)" title="Teorema Lagrange (teori grup) – Indonesian" lang="id" hreflang="id" data-title="Teorema Lagrange (teori grup)" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teorema_di_Lagrange_(teoria_dei_gruppi)" title="Teorema di Lagrange (teoria dei gruppi) – Italian" lang="it" hreflang="it" data-title="Teorema di Lagrange (teoria dei gruppi)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="משפט לגראנז' (תורת החבורות) – Hebrew" lang="he" hreflang="he" data-title="משפט לגראנז' (תורת החבורות)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B8%D0%B9%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC_(%D0%B1%D2%AF%D0%BB%D0%B3%D0%B8%D0%B9%D0%BD_%D0%BE%D0%BD%D0%BE%D0%BB)" title="Лагранжийн теорем (бүлгийн онол) – Mongolian" lang="mn" hreflang="mn" data-title="Лагранжийн теорем (бүлгийн онол)" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Lagrange_(groepentheorie)" title="Stelling van Lagrange (groepentheorie) – Dutch" lang="nl" hreflang="nl" data-title="Stelling van Lagrange (groepentheorie)" 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/%E3%83%A9%E3%82%B0%E3%83%A9%E3%83%B3%E3%82%B8%E3%83%A5%E3%81%AE%E5%AE%9A%E7%90%86_(%E7%BE%A4%E8%AB%96)" title="ラグランジュの定理 (群論) – Japanese" lang="ja" hreflang="ja" data-title="ラグランジュの定理 (群論)" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Teorema_%C3%ABd_Lagrange" title="Teorema ëd Lagrange – Piedmontese" lang="pms" hreflang="pms" data-title="Teorema ëd Lagrange" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Twierdzenie_Lagrange%E2%80%99a_(teoria_grup)" title="Twierdzenie Lagrange’a (teoria grup) – Polish" lang="pl" hreflang="pl" data-title="Twierdzenie Lagrange’a (teoria grup)" 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/Teorema_de_Lagrange_(teoria_dos_grupos)" title="Teorema de Lagrange (teoria dos grupos) – Portuguese" lang="pt" hreflang="pt" data-title="Teorema de Lagrange (teoria dos grupos)" 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/Teorema_lui_Lagrange_(teoria_grupurilor)" title="Teorema lui Lagrange (teoria grupurilor) – Romanian" lang="ro" hreflang="ro" data-title="Teorema lui Lagrange (teoria grupurilor)" 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%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF)" title="Теорема Лагранжа (теория групп) – Russian" lang="ru" hreflang="ru" data-title="Теорема Лагранжа (теория групп)" 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/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Lagrange's theorem (group theory)" 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/Lagrangeova_veta_(te%C3%B3ria_gr%C3%BAp)" title="Lagrangeova veta (teória grúp) – Slovak" lang="sk" hreflang="sk" data-title="Lagrangeova veta (teória grúp)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%BE%D0%B2%D0%B0_%D1%82%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0)" title="Лагранжова теорема (теорија група) – Serbian" lang="sr" hreflang="sr" data-title="Лагранжова теорема (теорија група)" 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/Lagrangeova_teorema_(teorija_grupa)" title="Lagrangeova teorema (teorija grupa) – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Lagrangeova teorema (teorija grupa)" 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/Lagrangen_indeksilause" title="Lagrangen indeksilause – Finnish" lang="fi" hreflang="fi" data-title="Lagrangen indeksilause" 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/Lagranges_sats" title="Lagranges sats – Swedish" lang="sv" hreflang="sv" data-title="Lagranges sats" 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%B2%E0%AE%BE%E0%AE%95%E0%AF%8D%E0%AE%B0%E0%AE%BE%E0%AE%9E%E0%AF%8D%E0%AE%9A%E0%AE%BF%E0%AE%AF%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%A4%E0%AF%87%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AE%AE%E0%AF%8D" title="லாக்ராஞ்சியின் தேற்றம் – Tamil" lang="ta" hreflang="ta" data-title="லாக்ராஞ்சியின் தேற்றம்" 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/Lagrange_teoremi_(grup_teorisi)" title="Lagrange teoremi (grup teorisi) – Turkish" lang="tr" hreflang="tr" data-title="Lagrange teoremi (grup teorisi)" 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%A2%D0%B5%D0%BE%D1%80%D0%B5%D0%BC%D0%B0_%D0%9B%D0%B0%D0%B3%D1%80%D0%B0%D0%BD%D0%B6%D0%B0_(%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B3%D1%80%D1%83%D0%BF)" title="Теорема Лагранжа (теорія груп) – Ukrainian" lang="uk" hreflang="uk" data-title="Теорема Лагранжа (теорія груп)" 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/%C4%90%E1%BB%8Bnh_l%C3%BD_Lagrange_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)" title="Định lý Lagrange (lý thuyết nhóm) – Vietnamese" lang="vi" hreflang="vi" data-title="Định lý Lagrange (lý thuyết nhóm)" 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 mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8B%89%E6%A0%BC%E6%9C%97%E6%97%A5%E5%AE%9A%E7%90%86_(%E7%BE%A4%E8%AB%96)" title="拉格朗日定理 (群論) – Chinese" lang="zh" hreflang="zh" data-title="拉格朗日定理 (群論)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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topics</a></li></ul></td> </tr></tbody></table></div></div></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)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> 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 class="mw-selflink selflink">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> <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>G 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> </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/34e4b343f070c08c1500541ce768b60f9b37bf9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; 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 H 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> </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/1b3bb21abe942aa9c0c63bae35a0c38905e1712c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }"></span>. There are four left cosets of H: H itself, 1+H, 2+H, and 3+H (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 G into equal-size, non-overlapping sets. Thus the <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> [G : H] is 4.</figcaption></figure> <p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> field of <a href="/wiki/Group_theory" title="Group theory">group theory</a>, <b>Lagrange's theorem</b> states that if H is a subgroup of any <a href="/wiki/Finite_group" title="Finite group">finite group</a> <span class="texhtml mvar" style="font-style:italic;">G</span>, then <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 |H|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |H|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c3c97e64ad558dcc09e9ff232ee0f4d447bac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.357ex; height:2.843ex;" alt="{\displaystyle |H|}"></span> is a divisor of <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|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |G|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8258bc41edeb87bfbc8cba8367f29838c0eddc1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.12ex; height:2.843ex;" alt="{\displaystyle |G|}"></span>, i.e. the <a href="/wiki/Order_of_a_group" class="mw-redirect" title="Order of a group">order</a> (number of elements) of every <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> H divides the order of group G. </p><p>The theorem is named after <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>. The following variant states that for a subgroup <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> of a finite 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 G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, not only is <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|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |G|/|H|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cd110506d8be74e70cb1dcb331f4971c099a26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.64ex; height:2.843ex;" alt="{\displaystyle |G|/|H|}"></span> an integer, but its value is the <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</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 [G: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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c19ed6f18e6db133b5a0257ecde8026808fd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle [G:H]}"></span>, defined as the number of left <a href="/wiki/Coset" title="Coset">cosets</a> of <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> in <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. </p> <style data-mw-deduplicate="TemplateStyles:r1110004140">.mw-parser-output .math_theorem{margin:1em 2em;padding:0.5em 1em 0.4em;border:1px solid #aaa;overflow:hidden}@media(max-width:500px){.mw-parser-output .math_theorem{margin:1em 0em;padding:0.5em 0.5em 0.4em}}</style><div class="math_theorem" style=""> <p><strong class="theorem-name">Lagrange's theorem</strong><span class="theoreme-tiret"> — </span>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>, then <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 \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>G</mi> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>G</mi> <mo>:</mo> <mi>H</mi> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mrow> <mo>|</mo> <mi>H</mi> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/642e10879449062a31d1276914ff0542f7c7faa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.41ex; height:2.843ex;" alt="{\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}"></span> </p> </div> <p>This variant holds even if <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is infinite, provided that <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|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |G|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8258bc41edeb87bfbc8cba8367f29838c0eddc1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.12ex; height:2.843ex;" alt="{\displaystyle |G|}"></span>, <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 |H|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |H|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31c3c97e64ad558dcc09e9ff232ee0f4d447bac1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.357ex; height:2.843ex;" alt="{\displaystyle |H|}"></span>, and <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]}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c19ed6f18e6db133b5a0257ecde8026808fd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle [G:H]}"></span> are interpreted as <a href="/wiki/Cardinal_number" title="Cardinal number">cardinal numbers</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange%27s_theorem_(group_theory)&action=edit&section=1" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The left <a href="/wiki/Coset" title="Coset">cosets</a> of <span class="texhtml mvar" style="font-style:italic;">H</span> in <span class="texhtml mvar" style="font-style:italic;">G</span> are the <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> of a certain <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on <span class="texhtml mvar" style="font-style:italic;">G</span>: specifically, call <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> in <span class="texhtml mvar" style="font-style:italic;">G</span> equivalent if there exists <span class="texhtml mvar" style="font-style:italic;">h</span> in <span class="texhtml mvar" style="font-style:italic;">H</span> such that <span class="texhtml"><i>x</i> = <i>yh</i></span>. Therefore, the set of left cosets forms a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of <span class="texhtml mvar" style="font-style:italic;">G</span>. Each left coset <span class="texhtml"><i>aH</i></span> has the same cardinality as <span class="texhtml mvar" style="font-style:italic;">H</span> because <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 x\mapsto ax}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>a</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto ax}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca6dfd2089f8884a4cdef73f987cb757062b7a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.503ex; height:1.843ex;" alt="{\displaystyle x\mapsto ax}"></span> defines a bijection <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 H\to aH}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">→</mo> <mi>a</mi> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\to aH}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06d7c55b8dea1d26c0d8160048d90761ed07cd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.971ex; height:2.176ex;" alt="{\displaystyle H\to aH}"></span> (the inverse is <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 y\mapsto a^{-1}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">↦</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\mapsto a^{-1}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a5c2325685e8dfc9b3203c259d9e7a25d29f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.488ex; height:3.009ex;" alt="{\displaystyle y\mapsto a^{-1}y}"></span>). The number of left cosets is the <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">index</a> <span class="texhtml">[<i>G</i> : <i>H</i>]</span>. By the previous three sentences, </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 \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mi>G</mi> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>G</mi> <mo>:</mo> <mi>H</mi> </mrow> <mo>]</mo> </mrow> <mo>⋅</mo> <mrow> <mo>|</mo> <mi>H</mi> <mo>|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/642e10879449062a31d1276914ff0542f7c7faa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.41ex; height:2.843ex;" alt="{\displaystyle \left|G\right|=\left[G:H\right]\cdot \left|H\right|.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Extension">Extension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange%27s_theorem_(group_theory)&action=edit&section=2" title="Edit section: Extension"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lagrange's theorem can be extended to the equation of indexes between three subgroups of <span class="texhtml mvar" style="font-style:italic;">G</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1110004140"><div class="math_theorem" style=""> <p><strong class="theorem-name">Extension of Lagrange's theorem</strong><span class="theoreme-tiret"> — </span>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> and <span class="texhtml mvar" style="font-style:italic;">K</span> is a subgroup of <span class="texhtml mvar" style="font-style:italic;">H</span>, then </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:K]=[G:H]\,[H:K].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo stretchy="false">[</mo> <mi>H</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:K]=[G:H]\,[H:K].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a79cd7e668c3e56bac480e737e939469973b9c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.738ex; height:2.843ex;" alt="{\displaystyle [G:K]=[G:H]\,[H:K].}"></span></dd></dl> </div> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>Let <span class="texhtml mvar" style="font-style:italic;">S</span> be a set of coset representatives for <span class="texhtml mvar" style="font-style:italic;">K</span> in <span class="texhtml mvar" style="font-style:italic;">H</span>, so <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 H=\bigsqcup _{s\in S}sK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mi>s</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=\bigsqcup _{s\in S}sK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/042b144c85a26fcfbae78e0ec4cd0fdef9358ceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.633ex; height:5.676ex;" alt="{\displaystyle H=\bigsqcup _{s\in S}sK}"></span> (disjoint union), and <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 |S|=[H:K]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi>H</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|=[H:K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec13f9e72163e3a482b0ded6b57112f030911db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.252ex; height:2.843ex;" alt="{\displaystyle |S|=[H:K]}"></span>. For any <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\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9f5ea7aea0b7a62b07eae139e7a5038ea5a120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle a\in G}"></span>, left-multiplication-by-<span class="texhtml mvar" style="font-style:italic;">a</span> is a bijection <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\to G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\to G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f4a9c67f1896ed706ac35c5e143d0726945ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:2.176ex;" alt="{\displaystyle G\to G}"></span>, so <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 aH=\bigsqcup _{s\in S}asK}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>H</mi> <mo>=</mo> <munder> <mo>⨆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <mi>a</mi> <mi>s</mi> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle aH=\bigsqcup _{s\in S}asK}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7bf02ac038e5a33cf982d1cabe9c1910f836c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:14.093ex; height:5.676ex;" alt="{\displaystyle aH=\bigsqcup _{s\in S}asK}"></span>. Thus each left coset of <span class="texhtml mvar" style="font-style:italic;">H</span> decomposes into <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 [H:K]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>H</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [H:K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4420311ded875925dd1959f6885c10ce31cc52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.36ex; height:2.843ex;" alt="{\displaystyle [H:K]}"></span> left cosets of <span class="texhtml mvar" style="font-style:italic;">K</span>. Since <span class="texhtml mvar" style="font-style:italic;">G</span> decomposes into <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]}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c19ed6f18e6db133b5a0257ecde8026808fd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle [G:H]}"></span> left cosets of <span class="texhtml mvar" style="font-style:italic;">H</span>, each of which decomposes into <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 [H:K]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>H</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [H:K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4420311ded875925dd1959f6885c10ce31cc52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.36ex; height:2.843ex;" alt="{\displaystyle [H:K]}"></span> left cosets of <span class="texhtml mvar" style="font-style:italic;">K</span>, the total number <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:K]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0c9e458289b60c3e2dd954679d0fc309ca2f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.124ex; height:2.843ex;" alt="{\displaystyle [G:K]}"></span> of left cosets of <span class="texhtml mvar" style="font-style:italic;">K</span> in <span class="texhtml mvar" style="font-style:italic;">G</span> is <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][H:K]}"> <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 stretchy="false">[</mo> <mi>H</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H][H:K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5559fd05e523f10bda5128eb89c368f533614c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.482ex; height:2.843ex;" alt="{\displaystyle [G:H][H:K]}"></span>. </p> </div> <p>If we take <span class="texhtml"><i>K</i> = {<i>e</i>}</span> (<span class="texhtml mvar" style="font-style:italic;">e</span> is the identity element of <span class="texhtml mvar" style="font-style:italic;">G</span>), then <span class="texhtml">[<i>G</i> : {<i>e</i>}] = |<i>G</i>|</span> and <span class="texhtml">[<i>H</i> : {<i>e</i>}] = |<i>H</i>|</span>. Therefore, we can recover the original equation <span class="texhtml">|<i>G</i>| = [<i>G</i> : <i>H</i>] |<i>H</i>|</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange%27s_theorem_(group_theory)&action=edit&section=3" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A consequence of the theorem is that the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order of any element</a> <span class="texhtml mvar" style="font-style:italic;">a</span> of a finite group (i.e. the smallest positive integer number <span class="texhtml mvar" style="font-style:italic;">k</span> with <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">a</span><sup><span class="texhtml mvar" style="font-style:italic;">k</span></sup> = <span class="texhtml mvar" style="font-style:italic;">e</span></span>, where <span class="texhtml mvar" style="font-style:italic;">e</span> is the identity element of the group) divides the order of that group, since the order of <span class="texhtml mvar" style="font-style:italic;">a</span> is equal to the order of the <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a> subgroup <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by <span class="texhtml mvar" style="font-style:italic;">a</span>. If the group has <span class="texhtml mvar" style="font-style:italic;">n</span> elements, it follows </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 \displaystyle a^{n}=e{\mbox{.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>.</mtext> </mstyle> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle a^{n}=e{\mbox{.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e029cdb02da84da9524ad585cdd66b3ea3f2149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.277ex; height:2.343ex;" alt="{\displaystyle \displaystyle a^{n}=e{\mbox{.}}}"></span></dd></dl> <p>This can be used to prove <a href="/wiki/Fermat%27s_little_theorem" title="Fermat's little theorem">Fermat's little theorem</a> and its generalization, <a href="/wiki/Euler%27s_theorem" title="Euler's theorem">Euler's theorem</a>. These special cases were known long before the general theorem was proved. </p><p>The theorem also shows that any group of prime order is cyclic and <a href="/wiki/Simple_group" title="Simple group">simple</a>, since the subgroup generated by any non-identity element must be the whole group itself. </p><p>Lagrange's theorem can also be used to show that there are infinitely many <a href="/wiki/Primes" class="mw-redirect" title="Primes">primes</a>: suppose there were a largest prime <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. Any prime divisor <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 q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> of the <a href="/wiki/Mersenne_number" class="mw-redirect" title="Mersenne number">Mersenne number</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 2^{p}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5c5977dbf385ba719fbb90f67b0a3d91e1da6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.224ex; height:2.509ex;" alt="{\displaystyle 2^{p}-1}"></span> satisfies <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^{p}\equiv 1{\pmod {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>≡</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mi>q</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{p}\equiv 1{\pmod {q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8714cffebb4f3d6788cd6509785c983455f2cb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.236ex; height:2.843ex;" alt="{\displaystyle 2^{p}\equiv 1{\pmod {q}}}"></span> (see <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>), meaning that the order of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> in the <a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative group</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} /q\mathbb {Z} )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0752dea7e1143fd2afefefa43c49735aaaf06cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}}"></span> is <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span>. By Lagrange's theorem, the order of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> must divide the order of <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} /q\mathbb {Z} )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0752dea7e1143fd2afefefa43c49735aaaf06cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.196ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}}"></span>, which is <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 q-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfc49678846b112cde021e3cb52d9b3b15decaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.072ex; height:2.509ex;" alt="{\displaystyle q-1}"></span>. So <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> divides <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 q-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bfc49678846b112cde021e3cb52d9b3b15decaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.072ex; height:2.509ex;" alt="{\displaystyle q-1}"></span>, giving <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 p<q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo><</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p<q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f86c7ea4068f76f93c6a2a92c849c27303c9ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.427ex; height:2.176ex;" alt="{\displaystyle p<q}"></span>, contradicting the assumption that <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is the largest prime.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Existence_of_subgroups_of_given_order">Existence of subgroups of given order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange%27s_theorem_(group_theory)&action=edit&section=4" title="Edit section: Existence of subgroups of given order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lagrange's theorem raises the converse question as to whether every divisor of the order of a group is the order of some subgroup. This does not hold in general: given a finite group <i>G</i> and a divisor <i>d</i> of |<i>G</i>|, there does not necessarily exist a subgroup of <i>G</i> with order <i>d</i>. The smallest example is <i>A</i><sub>4</sub> (the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> of degree 4), which has 12 elements but no subgroup of order 6. </p><p>A "Converse of Lagrange's Theorem" (CLT) group is a finite group with the property that for every divisor of the order of the group, there is a subgroup of that order. It is known that a CLT group must be <a href="/wiki/Solvable_group" title="Solvable group">solvable</a> and that every <a href="/wiki/Supersolvable_group" title="Supersolvable group">supersolvable group</a> is a CLT group. However, there exist solvable groups that are not CLT (for example, <i>A</i><sub>4</sub>) and CLT groups that are not supersolvable (for example, <i>S</i><sub>4</sub>, the symmetric group of degree 4). </p><p>There are partial converses to Lagrange's theorem. For general groups, <a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a> guarantees the existence of an element, and hence of a cyclic subgroup, of order any prime dividing the group order. <a href="/wiki/Sylow%27s_theorem" class="mw-redirect" title="Sylow's theorem">Sylow's theorem</a> extends this to the existence of a subgroup of order equal to the maximal power of any prime dividing the group order. For solvable groups, <a href="/wiki/Hall_subgroup#Hall's_theorem" title="Hall subgroup">Hall's theorems</a> assert the existence of a subgroup of order equal to any <a href="/wiki/Unitary_divisor" title="Unitary divisor">unitary divisor</a> of the group order (that is, a divisor coprime to its cofactor). </p> <div class="mw-heading mw-heading3"><h3 id="Counterexample_of_the_converse_of_Lagrange's_theorem"><span id="Counterexample_of_the_converse_of_Lagrange.27s_theorem"></span>Counterexample of the converse of Lagrange's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange%27s_theorem_(group_theory)&action=edit&section=5" title="Edit section: Counterexample of the converse of Lagrange's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The converse of Lagrange's theorem states that if <span class="texhtml mvar" style="font-style:italic;">d</span> is a <a href="/wiki/Divisor" title="Divisor">divisor</a> of the order of a group <span class="texhtml mvar" style="font-style:italic;">G</span>, then there exists a subgroup <span class="texhtml mvar" style="font-style:italic;">H</span> where <span class="texhtml">|<i>H</i>| = <i>d</i></span>. </p><p>We will examine the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> <span class="texhtml"><i>A</i><sub>4</sub></span>, the set of even <a href="/wiki/Permutation_group" title="Permutation group">permutations</a> as the subgroup of the <a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> <span class="texhtml"><i>S</i><sub>4</sub></span>. </p> <dl><dd><span class="nowrap"><span class="texhtml"><i>A</i><sub>4</sub> = {<i>e</i>, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 2 3), (1 3 2), (1 2 4), (1 4 2), (1 3 4), (1 4 3), (2 3 4), (2 4 3)}</span>.</span></dd></dl> <p><span class="texhtml">|<i>A</i><sub>4</sub>| = 12</span> so the divisors are <span class="texhtml">1, 2, 3, 4, 6, 12</span>. Assume to the contrary that there exists a subgroup <span class="texhtml mvar" style="font-style:italic;">H</span> in <span class="texhtml"><i>A</i><sub>4</sub></span> with <span class="texhtml">|<i>H</i>| = 6</span>. </p><p>Let <span class="texhtml mvar" style="font-style:italic;">V</span> be the <a href="/wiki/Cyclic_group" title="Cyclic group">non-cyclic</a> subgroup of <span class="texhtml"><i>A</i><sub>4</sub></span> called the <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a>. </p> <dl><dd><span class="texhtml"><i>V</i> = {<i>e</i>, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}</span>.</dd></dl> <p>Let <span class="texhtml"><i>K</i> = <i>H</i> ⋂ <i>V</i></span>. Since both <span class="texhtml mvar" style="font-style:italic;">H</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> are subgroups of <span class="texhtml"><i>A</i><sub>4</sub></span>, <span class="texhtml mvar" style="font-style:italic;">K</span> is also a subgroup of <span class="texhtml"><i>A</i><sub>4</sub></span>. </p><p>From Lagrange's theorem, the order of <span class="texhtml mvar" style="font-style:italic;">K</span> must divide both <span class="texhtml">6</span> and <span class="texhtml">4</span>, the orders of <span class="texhtml mvar" style="font-style:italic;">H</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> respectively. The only two positive integers that divide both <span class="texhtml">6</span> and <span class="texhtml">4</span> are <span class="texhtml">1</span> and <span class="texhtml">2</span>. So <span class="texhtml">|<i>K</i>| = 1</span> or <span class="texhtml">2</span>. </p><p>Assume <span class="texhtml">|<i>K</i>| = 1</span>, then <span class="texhtml"><i>K</i> = {<i>e</i>}</span>. If <span class="texhtml mvar" style="font-style:italic;">H</span> does not share any elements with <span class="texhtml mvar" style="font-style:italic;">V</span>, then the 5 elements in <span class="texhtml mvar" style="font-style:italic;">H</span> besides the <a href="/wiki/Identity_element" title="Identity element">Identity element</a> <span class="texhtml mvar" style="font-style:italic;">e</span> must be of the form <span class="texhtml">(<i>a b c</i>)</span> where <span class="texhtml"><i>a, b, c</i></span> are distinct elements in <span class="texhtml">{1, 2, 3, 4}</span>. </p><p>Since any element of the form <span class="texhtml">(<i>a b c</i>)</span> squared is <span class="texhtml">(<i>a c b</i>)</span>, and <span class="texhtml">(<i>a b c</i>)(<i>a c b</i>) = <i>e</i></span>, any element of <span class="texhtml mvar" style="font-style:italic;">H</span> in the form <span class="texhtml">(<i>a b c</i>)</span> must be paired with its inverse. Specifically, the remaining 5 elements of <span class="texhtml mvar" style="font-style:italic;">H</span> must come from distinct pairs of elements in <span class="texhtml"><i>A</i><sub>4</sub></span> that are not in <span class="texhtml mvar" style="font-style:italic;">V</span>. This is impossible since pairs of elements must be even and cannot total up to 5 elements. Thus, the assumptions that <span class="texhtml">|<i>K</i>| = 1</span> is wrong, so <span class="texhtml">|<i>K</i>| = 2</span>. </p><p>Then, <span class="texhtml"><i>K</i> = {<i>e</i>, <i>v</i>}</span> where <span class="texhtml"><i>v</i> ∈ <i>V</i></span>, <span class="texhtml mvar" style="font-style:italic;">v</span> must be in the form <span class="texhtml">(<i>a b</i>)(<i>c d</i>)</span> where <span class="texhtml mvar" style="font-style:italic;">a, b, c, d</span> are distinct elements of <span class="texhtml">{1, 2, 3, 4}</span>. The other four elements in <span class="texhtml mvar" style="font-style:italic;">H</span> are cycles of length 3. </p><p>Note that the cosets <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generated</a> by a subgroup of a group form a partition of the group. The cosets generated by a specific subgroup are either identical to each other or <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a>. The index of a subgroup in a group <span class="texhtml">[<i>A</i><sub>4</sub> : <i>H</i>] = |<i>A</i><sub>4</sub>|/|<i>H</i>|</span> is the number of cosets generated by that subgroup. Since <span class="texhtml">|<i>A</i><sub>4</sub>| = 12</span> and <span class="texhtml">|<i>H</i>| = 6</span>, <span class="texhtml mvar" style="font-style:italic;">H</span> will generate two left cosets, one that is equal to <span class="texhtml mvar" style="font-style:italic;">H</span> and another, <span class="texhtml mvar" style="font-style:italic;">gH</span>, that is of length 6 and includes all the elements in <span class="texhtml"><i>A</i><sub>4</sub></span> not in <span class="texhtml mvar" style="font-style:italic;">H</span>. </p><p>Since there are only 2 distinct cosets generated by <span class="texhtml mvar" style="font-style:italic;">H</span>, then <span class="texhtml mvar" style="font-style:italic;">H</span> must be normal. Because of that, <span class="texhtml"><i>H</i> = <i>gHg</i><sup>−1</sup> (∀<i>g</i> ∈ <i>A</i><sub>4</sub>)</span>. In particular, this is true for <span class="texhtml"><i>g</i> = (<i>a b c</i>) ∈ <i>A</i><sub>4</sub></span>. Since <span class="texhtml"><i>H</i> = <i>gHg</i><sup>−1</sup>, <i>gvg</i><sup>−1</sup> ∈ <i>H</i></span>. </p><p>Without loss of generality, assume that <span class="texhtml"><i>a</i> = 1</span>, <span class="texhtml"><i>b</i> = 2</span>, <span class="texhtml"><i>c</i> = 3</span>, <span class="texhtml"><i>d</i> = 4</span>. Then <span class="texhtml"><i>g</i> = (1 2 3)</span>, <span class="texhtml"><i>v</i> = (1 2)(3 4)</span>, <span class="texhtml"><i>g</i><sup>−1</sup> = (1 3 2)</span>, <span class="texhtml"><i>gv</i> = (1 3 4)</span>, <span class="texhtml"><i>gvg</i><sup>−1</sup> = (1 4)(2 3)</span>. Transforming back, we get <span class="texhtml"><i>gvg</i><sup>−1</sup> = (<i>a</i> <i>d</i>)(<i>b</i> <i>c</i>)</span>. Because <span class="texhtml mvar" style="font-style:italic;">V</span> contains all disjoint transpositions in <span class="texhtml"><i>A</i><sub>4</sub></span>, <span class="texhtml"><i>gvg</i><sup>−1</sup> ∈ <i>V</i></span>. Hence, <span class="texhtml"><i>gvg</i><sup>−1</sup> ∈ <i>H</i> ⋂ <i>V</i> = <i>K</i></span>. </p><p>Since <span class="texhtml"><i>gvg</i><sup>−1</sup> ≠ <i>v</i></span>, we have demonstrated that there is a third element in <span class="texhtml mvar" style="font-style:italic;">K</span>. But earlier we assumed that <span class="texhtml">|<i>K</i>| = 2</span>, so we have a contradiction. </p><p>Therefore, our original assumption that there is a subgroup of order 6 is not true and consequently there is no subgroup of order 6 in <span class="texhtml"><i>A</i><sub>4</sub></span> and the converse of Lagrange's theorem is not necessarily true. <a href="/wiki/Q.E.D." title="Q.E.D.">Q.E.D.</a> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lagrange%27s_theorem_(group_theory)&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lagrange himself did not prove the theorem in its general form. He stated, in his article <i>Réflexions sur la résolution algébrique des équations</i>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> that if a polynomial in <span class="texhtml mvar" style="font-style:italic;">n</span> variables has its variables permuted in all <span class="texhtml"><i>n</i>!</span> ways, the number of different polynomials that are obtained is always a factor of <span class="texhtml"><i>n</i>!</span>. (For example, if the variables <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span>, and <span class="texhtml mvar" style="font-style:italic;">z</span> are permuted in all 6 possible ways in the polynomial <span class="texhtml"><i>x</i> + <i>y</i> − <i>z</i></span> then we get a total of 3 different polynomials: <span class="texhtml"><i>x</i> + <i>y</i> − <i>z</i></span>, <span class="texhtml"><i>x</i> + <i>z</i> − <i>y</i></span>, and <span class="texhtml"><i>y</i> + <i>z</i> − <i>x</i></span>. Note that 3 is a factor of 6.) The number of such polynomials is the index in the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> <span class="texhtml"><i>S</i><sub>n</sub></span> of the subgroup <span class="texhtml mvar" style="font-style:italic;"><i>H</i></span> of permutations that preserve the polynomial. (For the example of <span class="texhtml"><i>x</i> + <i>y</i> − <i>z</i></span>, the subgroup <span class="texhtml mvar" style="font-style:italic;"><i>H</i></span> in <span class="texhtml"><i>S</i><sub>3</sub></span> contains the identity and the transposition <span class="texhtml">(<i>x y</i>)</span>.) So the size of <span class="texhtml mvar" style="font-style:italic;"><i>H</i></span> divides <span class="texhtml"><i>n</i>!</span>. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name. </p><p>In his <i><a href="/wiki/Disquisitiones_Arithmeticae" title="Disquisitiones Arithmeticae">Disquisitiones Arithmeticae</a></i> in 1801, <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> proved Lagrange's theorem for the special case of <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} /p\mathbb {Z} )^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa91bab9695e269c49abee9b4b31fe8a1211a8e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.296ex; height:2.843ex;" alt="{\displaystyle (\mathbb {Z} /p\mathbb {Z} )^{*}}"></span>, the multiplicative group of nonzero integers <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> <span class="texhtml mvar" style="font-style:italic;">p</span>, where <span class="texhtml mvar" style="font-style:italic;">p</span> is a prime.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> In 1844, <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> proved Lagrange's theorem for the symmetric group <span class="texhtml"><i>S</i><sub>n</sub></span>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a> finally proved Lagrange's theorem for the case of any <a href="/wiki/Permutation_group" title="Permutation group">permutation group</a> in 1861.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <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=Lagrange%27s_theorem_(group_theory)&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><span class="citation mathworld" id="Reference-Mathworld-Lagrange's_Group_Theorem"><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="CITEREFWeisstein" class="citation web cs2">Bray, Nicolas, <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/LagrangesGroupTheorem.html">"Lagrange's Group Theorem"</a>, <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Lagrange%27s+Group+Theorem&rft.au=Bray%2C+Nicolas&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FLagrangesGroupTheorem.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAignerZiegler2018" class="citation cs2"><a href="/wiki/Martin_Aigner" title="Martin Aigner">Aigner, Martin</a>; <a href="/wiki/G%C3%BCnter_M._Ziegler" title="Günter M. Ziegler">Ziegler, Günter M.</a> (2018), "Chapter 1", <i><a href="/wiki/Proofs_from_THE_BOOK" title="Proofs from THE BOOK">Proofs from THE BOOK</a></i> (Revised and enlarged sixth ed.), Berlin: Springer, pp. <span class="nowrap">3–</span>8, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-662-57264-1" title="Special:BookSources/978-3-662-57264-1"><bdi>978-3-662-57264-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1&rft.btitle=Proofs+from+THE+BOOK&rft.place=Berlin&rft.pages=%3Cspan+class%3D%22nowrap%22%3E3-%3C%2Fspan%3E8&rft.edition=Revised+and+enlarged+sixth&rft.pub=Springer&rft.date=2018&rft.isbn=978-3-662-57264-1&rft.aulast=Aigner&rft.aufirst=Martin&rft.au=Ziegler%2C+G%C3%BCnter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLagrange1771" class="citation cs2"><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange, Joseph-Louis</a> (1771), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_-U_AAAAYAAJ&pg=PA138">"Suite des réflexions sur la résolution algébrique des équations. Section troisieme. De la résolution des équations du cinquieme degré & des degrés ultérieurs."</a> [Series of reflections on the algebraic solution of equations. Third section. On the solution of equations of the fifth degree & higher degrees], <i>Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin</i>: <span class="nowrap">138–</span>254</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nouveaux+M%C3%A9moires+de+l%27Acad%C3%A9mie+Royale+des+Sciences+et+Belles-Lettres+de+Berlin&rft.atitle=Suite+des+r%C3%A9flexions+sur+la+r%C3%A9solution+alg%C3%A9brique+des+%C3%A9quations.+Section+troisieme.+De+la+r%C3%A9solution+des+%C3%A9quations+du+cinquieme+degr%C3%A9+%26+des+degr%C3%A9s+ult%C3%A9rieurs.&rft.pages=%3Cspan+class%3D%22nowrap%22%3E138-%3C%2Fspan%3E254&rft.date=1771&rft.aulast=Lagrange&rft.aufirst=Joseph-Louis&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_-U_AAAAYAAJ%26pg%3DPA138&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span> ; see especially <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_-U_AAAAYAAJ&pg=PA202">pages 202-203.</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1801" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss, Carl Friedrich</a> (1801), <i>Disquisitiones Arithmeticae</i> (in Latin), Leipzig (Lipsia): G. Fleischer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Disquisitiones+Arithmeticae&rft.place=Leipzig+%28Lipsia%29&rft.pub=G.+Fleischer&rft.date=1801&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span>, <a rel="nofollow" class="external text" href="http://babel.hathitrust.org/cgi/pt?id=nyp.33433070725894;view=1up;seq=63">pp. 41-45, Art. 45-49.</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a>, <i>§VI. — Sur les dérivées d'une ou de plusieurs substitutions, et sur les systèmes de substitutions conjuguées</i> [On the products of one or several permutations, and on systems of conjugate permutations] of: <i>"Mémoire sur les arrangements que l'on peut former avec des lettres données, et sur les permutations ou substitutions à l'aide desquelles on passe d'un arrangement à un autre"</i> [Memoir on the arrangements that one can form with given letters, and on the permutations or substitutions by means of which one passes from one arrangement to another] in: <i>Exercises d'analyse et de physique mathématique</i> [Exercises in analysis and mathematical physics], vol. 3 (Paris, France: Bachelier, 1844), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-c3fxufDQVEC&pg=PA183">pp. 183-185.</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJordan1861" class="citation cs2"><a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan, Camille</a> (1861), <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k433691p/f118.image.langEN">"Mémoire sur le numbre des valeurs des fonctions"</a> [Memoir on the number of values of functions], <i>Journal de l'École Polytechnique</i>, <b>22</b>: <span class="nowrap">113–</span>194</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+l%27%C3%89cole+Polytechnique&rft.atitle=M%C3%A9moire+sur+le+numbre+des+valeurs+des+fonctions&rft.volume=22&rft.pages=%3Cspan+class%3D%22nowrap%22%3E113-%3C%2Fspan%3E194&rft.date=1861&rft.aulast=Jordan&rft.aufirst=Camille&rft_id=http%3A%2F%2Fgallica.bnf.fr%2Fark%3A%2F12148%2Fbpt6k433691p%2Ff118.image.langEN&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span> Jordan's generalization of Lagrange's theorem appears on <a rel="nofollow" class="external text" href="http://gallica.bnf.fr/ark:/12148/bpt6k433691p/f171.image.r=Lagrange.langEN">page 166.</a></span> </li> </ol></div></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=Lagrange%27s_theorem_(group_theory)&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBray1968" class="citation cs2">Bray, Henry G. (1968), "A note on CLT groups", <i>Pacific J. Math.</i>, <b>27</b> (2): <span class="nowrap">229–</span>231, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2140%2Fpjm.1968.27.229">10.2140/pjm.1968.27.229</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pacific+J.+Math.&rft.atitle=A+note+on+CLT+groups&rft.volume=27&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E231&rft.date=1968&rft_id=info%3Adoi%2F10.2140%2Fpjm.1968.27.229&rft.aulast=Bray&rft.aufirst=Henry+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallian2006" class="citation cs2">Gallian, Joseph (2006), <i>Contemporary Abstract Algebra</i> (6th ed.), Boston: Houghton Mifflin, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-618-51471-7" title="Special:BookSources/978-0-618-51471-7"><bdi>978-0-618-51471-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Abstract+Algebra&rft.place=Boston&rft.edition=6th&rft.pub=Houghton+Mifflin&rft.date=2006&rft.isbn=978-0-618-51471-7&rft.aulast=Gallian&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDummitFoote2004" class="citation cs2">Dummit, David S.; Foote, Richard M. (2004), <i>Abstract algebra</i> (3rd ed.), New York: <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-43334-7" title="Special:BookSources/978-0-471-43334-7"><bdi>978-0-471-43334-7</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2286236">2286236</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=New+York&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=2004&rft.isbn=978-0-471-43334-7&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2286236%23id-name%3DMR&rft.aulast=Dummit&rft.aufirst=David+S.&rft.au=Foote%2C+Richard+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALagrange%27s+theorem+%28group+theory%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRoth2001" class="citation cs2">Roth, Richard R. (2001), "A History of Lagrange's Theorem on Groups", <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>, <b>74</b> (2): <span class="nowrap">99–</span>108, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690624">10.2307/2690624</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690624">2690624</a></cite><span 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