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<span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Permutation_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Permutation_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Permutation groups</span> </div> </a> <ul id="toc-Permutation_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cyclic_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cyclic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Cyclic groups</span> </div> </a> <ul id="toc-Cyclic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Finite_abelian_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_abelian_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Finite abelian groups</span> </div> </a> <ul id="toc-Finite_abelian_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Groups_of_Lie_type" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Groups_of_Lie_type"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Groups of Lie type</span> </div> </a> <ul id="toc-Groups_of_Lie_type-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Main_theorems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Main_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Main theorems</span> </div> </a> <button aria-controls="toc-Main_theorems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Main theorems subsection</span> </button> <ul id="toc-Main_theorems-sublist" class="vector-toc-list"> <li id="toc-Lagrange's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lagrange's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Lagrange's theorem</span> </div> </a> <ul id="toc-Lagrange's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sylow_theorems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sylow_theorems"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Sylow theorems</span> </div> </a> <ul id="toc-Sylow_theorems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cayley's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cayley's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Cayley's theorem</span> </div> </a> <ul id="toc-Cayley's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Burnside's_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Burnside's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Burnside's theorem</span> </div> </a> <ul id="toc-Burnside's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feit–Thompson_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feit–Thompson_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Feit–Thompson theorem</span> </div> </a> <ul id="toc-Feit–Thompson_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_of_finite_simple_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classification_of_finite_simple_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.6</span> <span>Classification of finite simple groups</span> </div> </a> <ul id="toc-Classification_of_finite_simple_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Number_of_groups_of_a_given_order" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Number_of_groups_of_a_given_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Number of groups of a given order</span> </div> </a> <button aria-controls="toc-Number_of_groups_of_a_given_order-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Number of groups of a given order subsection</span> </button> <ul id="toc-Number_of_groups_of_a_given_order-sublist" class="vector-toc-list"> <li id="toc-Table_of_distinct_groups_of_order_n" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Table_of_distinct_groups_of_order_n"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Table of distinct groups of order <i>n</i></span> </div> </a> <ul id="toc-Table_of_distinct_groups_of_order_n-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">6</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Finite group</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BD%D0%B5%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Канечная група – Belarusian" lang="be" hreflang="be" data-title="Канечная група" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D1%80%D0%B0%D0%B9%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Крайна група – Bulgarian" lang="bg" hreflang="bg" data-title="Крайна група" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_finit" title="Grup finit – Catalan" lang="ca" hreflang="ca" data-title="Grup finit" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Endliche_Gruppe" title="Endliche Gruppe – German" lang="de" hreflang="de" data-title="Endliche Gruppe" 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/Grupo_finito" title="Grupo finito – Spanish" lang="es" hreflang="es" data-title="Grupo finito" 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/%DA%AF%D8%B1%D9%88%D9%87_%D9%85%D8%AA%D9%86%D8%A7%D9%87%DB%8C" title="گروه متناهی – Persian" lang="fa" hreflang="fa" data-title="گروه متناهی" 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/Groupe_fini" title="Groupe fini – French" lang="fr" hreflang="fr" data-title="Groupe fini" 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/%EC%9C%A0%ED%95%9C%EA%B5%B0" title="유한군 – Korean" lang="ko" hreflang="ko" data-title="유한군" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_hingga" title="Grup hingga – Indonesian" lang="id" hreflang="id" data-title="Grup hingga" 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/Gruppo_finito" title="Gruppo finito – Italian" lang="it" hreflang="it" data-title="Gruppo finito" 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%97%D7%91%D7%95%D7%A8%D7%94_%D7%A1%D7%95%D7%A4%D7%99%D7%AA" title="חבורה סופית – Hebrew" lang="he" hreflang="he" data-title="חבורה סופית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/V%C3%A9ges_csoport" title="Véges csoport – Hungarian" lang="hu" hreflang="hu" data-title="Véges csoport" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Eindige_groep" title="Eindige groep – Dutch" lang="nl" hreflang="nl" data-title="Eindige groep" 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/%E6%9C%89%E9%99%90%E7%BE%A4" title="有限群 – Japanese" lang="ja" hreflang="ja" data-title="有限群" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_finito" title="Grupo finito – Portuguese" lang="pt" hreflang="pt" data-title="Grupo finito" 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/Grup_finit" title="Grup finit – Romanian" lang="ro" hreflang="ro" data-title="Grup finit" 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%9A%D0%BE%D0%BD%D0%B5%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Конечная группа – Russian" lang="ru" hreflang="ru" data-title="Конечная группа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/%C3%84%C3%A4rellinen_ryhm%C3%A4" title="Äärellinen ryhmä – Finnish" lang="fi" hreflang="fi" data-title="Äärellinen ryhmä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%81%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AF%81%E0%AE%B1%E0%AF%81_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="முடிவுறு குலம் – Tamil" lang="ta" hreflang="ta" data-title="முடிவுறு குலம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BA%D1%96%D0%BD%D1%87%D0%B5%D0%BD%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Скінченна група – Ukrainian" lang="uk" hreflang="uk" data-title="Скінченна група" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_h%E1%BB%AFu_h%E1%BA%A1n" title="Nhóm hữu hạn – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm hữu hạn" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%9C%89%E9%99%90%E7%BE%A3" title="有限羣 – Cantonese" lang="yue" hreflang="yue" data-title="有限羣" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%9C%89%E9%99%90%E7%BE%A4" title="有限群 – Chinese" lang="zh" hreflang="zh" data-title="有限群" data-language-autonym="中文" data-language-local-name="Chinese" 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class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a class="mw-selflink selflink">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, a <b>finite group</b> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> whose <a href="/wiki/Underlying_set" class="mw-redirect" title="Underlying set">underlying set</a> is <a href="/wiki/Finite_set" title="Finite set">finite</a>. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. Important examples of finite groups include <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a> and <a href="/wiki/Permutation_group" title="Permutation group">permutation groups</a>. </p><p>The study of finite groups has been an integral part of <a href="/wiki/Group_theory" title="Group theory">group theory</a> since it arose in the 19th century. One major area of study has been classification: the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> (those with no nontrivial <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>) was completed in 2004. </p> <meta property="mw:PageProp/toc" /> <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=Finite_group&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the <a href="/wiki/Local_analysis" title="Local analysis">local theory</a> of finite groups and the theory of <a href="/wiki/Solvable_group" title="Solvable group">solvable</a> and <a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent groups</a>.<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><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> As a consequence, the complete <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> was achieved, meaning that all those <a href="/wiki/Simple_group" title="Simple group">simple groups</a> from which all finite groups can be built are now known. </p><p>During the second half of the twentieth century, mathematicians such as <a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Chevalley</a> and <a href="/wiki/Robert_Steinberg" title="Robert Steinberg">Steinberg</a> also increased our understanding of finite analogs of <a href="/wiki/Classical_groups" class="mw-redirect" title="Classical groups">classical groups</a>, and other related groups. One such family of groups is the family of <a href="/wiki/General_linear_group" title="General linear group">general linear groups</a> over <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. </p><p>Finite groups often occur when considering <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>, which may be viewed as dealing with "<a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetry</a>", is strongly influenced by the associated <a href="/wiki/Weyl_group" title="Weyl group">Weyl groups</a>. These are finite groups generated by reflections which act on a finite-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. The properties of finite groups can thus play a role in subjects such as <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Permutation_groups">Permutation groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=3" title="Edit section: Permutation groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Permutation_group" title="Permutation group">Permutation group</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_4;_Cayley_graph_4,9.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Symmetric_group_4%3B_Cayley_graph_4%2C9.svg/320px-Symmetric_group_4%3B_Cayley_graph_4%2C9.svg.png" decoding="async" width="320" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Symmetric_group_4%3B_Cayley_graph_4%2C9.svg/480px-Symmetric_group_4%3B_Cayley_graph_4%2C9.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Symmetric_group_4%3B_Cayley_graph_4%2C9.svg/640px-Symmetric_group_4%3B_Cayley_graph_4%2C9.svg.png 2x" data-file-width="812" data-file-height="812" /></a><figcaption>A <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a> of the symmetric group <a href="https://en.wikiversity.org/wiki/Symmetric_group_S4" class="extiw" title="v:Symmetric group S4">S<sub>4</sub></a></figcaption></figure> <p>The <b>symmetric group</b> S<sub><i>n</i></sub> on a <a href="/wiki/Finite_set" title="Finite set">finite set</a> of <i>n</i> symbols is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> whose elements are all the <a href="/wiki/Permutations" class="mw-redirect" title="Permutations">permutations</a> of the <i>n</i> symbols, and whose <a href="/wiki/Group_operation" class="mw-redirect" title="Group operation">group operation</a> is the <a href="/wiki/Function_composition" title="Function composition">composition</a> of such permutations, which are treated as <a href="/wiki/Bijection" title="Bijection">bijective functions</a> from the set of symbols to itself.<sup id="cite_ref-Jacobson-def_4-0" class="reference"><a href="#cite_note-Jacobson-def-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Since there are <i>n</i>! (<i>n</i> <a href="/wiki/Factorial" title="Factorial">factorial</a>) possible permutations of a set of <i>n</i> symbols, it follows that the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> (the number of elements) of the symmetric group S<sub><i>n</i></sub> is <i>n</i>!. </p> <div class="mw-heading mw-heading3"><h3 id="Cyclic_groups">Cyclic groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=4" title="Edit section: Cyclic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a></div> <p>A cyclic group Z<sub><i>n</i></sub> is a group all of whose elements are powers of a particular element <i>a</i> where <span class="nowrap"><i>a</i><span style="padding-left:0.12em;"><sup><i>n</i></sup></span> = <i>a</i><span style="padding-left:0.12em;"><sup>0</sup></span> = e</span>, the identity. A typical realization of this group is as the complex <a href="/wiki/Root_of_unity" title="Root of unity"><span style="white-space:nowrap"><i>n</i><span style="margin-left:0.12em">th</span></span> roots of unity</a>. Sending <i>a</i> to a <a href="/wiki/Primitive_root_of_unity" class="mw-redirect" title="Primitive root of unity">primitive root of unity</a> gives an isomorphism between the two. This can be done with any finite cyclic group. </p> <div class="mw-heading mw-heading3"><h3 id="Finite_abelian_groups">Finite abelian groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=5" title="Edit section: Finite abelian groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_abelian_group" class="mw-redirect" title="Finite abelian group">Finite abelian group</a></div> <p>An <b><a href="/wiki/Abelian_group" title="Abelian group">abelian group</a></b>, also called a <b>commutative group</b>, is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> in which the result of applying the group <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> to two group elements does not depend on their order (the axiom of <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>). They are named after <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>.<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>An arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of <a href="/wiki/Georg_Frobenius" class="mw-redirect" title="Georg Frobenius">Georg Frobenius</a> and <a href="/wiki/Ludwig_Stickelberger" title="Ludwig Stickelberger">Ludwig Stickelberger</a> and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Groups_of_Lie_type">Groups of Lie type</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=6" title="Edit section: Groups of Lie type"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Group of Lie type</a></div> <p>A <b>group of Lie type</b> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> closely related to the group <i>G</i>(<i>k</i>) of rational points of a reductive <a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">linear algebraic group</a> <i>G</i> with values in the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>k</i>. Finite groups of Lie type give the bulk of nonabelian <a href="/wiki/Finite_simple_groups" class="mw-redirect" title="Finite simple groups">finite simple groups</a>. Special cases include the <a href="/wiki/Classical_groups" class="mw-redirect" title="Classical groups">classical groups</a>, the <a href="/wiki/Chevalley_groups" class="mw-redirect" title="Chevalley groups">Chevalley groups</a>, the Steinberg groups, and the Suzuki–Ree groups. </p><p>Finite groups of Lie type were among the first groups to be considered in mathematics, after <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a>, <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric</a> and <a href="/wiki/Alternating_group" title="Alternating group">alternating</a> groups, with the <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear groups</a> over prime finite fields, PSL(2, <i>p</i>) being constructed by <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> in the 1830s. The systematic exploration of finite groups of Lie type started with <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a>'s theorem that the <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear group</a> PSL(2, <i>q</i>) is simple for <i>q</i> ≠ 2, 3. This theorem generalizes to projective groups of higher dimensions and gives an important infinite family PSL(<i>n</i>, <i>q</i>) of <a href="/wiki/Finite_simple_groups" class="mw-redirect" title="Finite simple groups">finite simple groups</a>. Other classical groups were studied by <a href="/wiki/Leonard_Dickson" class="mw-redirect" title="Leonard Dickson">Leonard Dickson</a> in the beginning of 20th century. In the 1950s <a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a> realized that after an appropriate reformulation, many theorems about <a href="/wiki/Semisimple_Lie_group" class="mw-redirect" title="Semisimple Lie group">semisimple Lie groups</a> admit analogues for algebraic groups over an arbitrary field <i>k</i>, leading to construction of what are now called <i>Chevalley groups</i>. Moreover, as in the case of compact simple Lie groups, the corresponding groups turned out to be almost simple as abstract groups (<i>Tits simplicity theorem</i>). Although it was known since 19th century that other finite simple groups exist (for example, <a href="/wiki/Mathieu_groups" class="mw-redirect" title="Mathieu groups">Mathieu groups</a>), gradually a belief formed that nearly all finite simple groups can be accounted for by appropriate extensions of Chevalley's construction, together with cyclic and alternating groups. Moreover, the exceptions, the <a href="/wiki/Sporadic_groups" class="mw-redirect" title="Sporadic groups">sporadic groups</a>, share many properties with the finite groups of Lie type, and in particular, can be constructed and characterized based on their <i>geometry</i> in the sense of Tits. </p><p>The belief has now become a theorem – the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>. Inspection of the list of finite simple groups shows that groups of Lie type over a <a href="/wiki/Finite_field" title="Finite field">finite field</a> include all the finite simple groups other than the cyclic groups, the alternating groups, the <a href="/wiki/Tits_group" title="Tits group">Tits group</a>, and the 26 <a href="/wiki/Sporadic_simple_group" class="mw-redirect" title="Sporadic simple group">sporadic simple groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Main_theorems">Main theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=7" title="Edit section: Main theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Lagrange's_theorem"><span id="Lagrange.27s_theorem"></span>Lagrange's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=8" title="Edit section: Lagrange's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem (group theory)</a></div> <p>For any finite group <i>G</i>, the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> (number of elements) of every <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> <i>H</i> of <i>G</i> divides the order of <i>G</i>. The theorem is named after <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Sylow_theorems">Sylow theorems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=9" title="Edit section: Sylow theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></div> <p>This provides a partial converse to Lagrange's theorem giving information about how many subgroups of a given order are contained in <i>G</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Cayley's_theorem"><span id="Cayley.27s_theorem"></span>Cayley's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=10" title="Edit section: Cayley's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a></div> <p><b>Cayley's theorem</b>, named in honour of <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a>, states that every <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> acting on <i>G</i>.<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> This can be understood as an example of the <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> of <i>G</i> on the elements of <i>G</i>.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Burnside's_theorem"><span id="Burnside.27s_theorem"></span>Burnside's theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=11" title="Edit section: Burnside's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Burnside%27s_theorem" title="Burnside's theorem">Burnside's theorem</a></div> <p><b>Burnside's theorem</b> in <a href="/wiki/Group_theory" title="Group theory">group theory</a> states that if <i>G</i> is a finite group of <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> <i>p</i><sup><i>a</i></sup><i>q</i><sup><i>b</i></sup>, where <i>p</i> and <i>q</i> are <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, and <i>a</i> and <i>b</i> are <a href="/wiki/Negative_and_positive_numbers" class="mw-redirect" title="Negative and positive numbers">non-negative</a> <a href="/wiki/Integer" title="Integer">integers</a>, then <i>G</i> is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a>. Hence each non-Abelian <a href="/wiki/Finite_simple_group" class="mw-redirect" title="Finite simple group">finite simple group</a> has order divisible by at least three distinct primes. </p> <div class="mw-heading mw-heading3"><h3 id="Feit–Thompson_theorem"><span id="Feit.E2.80.93Thompson_theorem"></span>Feit–Thompson theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=12" title="Edit section: Feit–Thompson theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <b>Feit–Thompson theorem</b>, or <b>odd order theorem</b>, states that every finite <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> of odd <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a>. It was proved by <a href="/wiki/Walter_Feit" title="Walter Feit">Walter Feit</a> and <a href="/wiki/John_Griggs_Thompson" class="mw-redirect" title="John Griggs Thompson">John Griggs Thompson</a> (<a href="#CITEREFFeitThompson1962">1962</a>, <a href="#CITEREFFeitThompson1963">1963</a>) </p> <div class="mw-heading mw-heading3"><h3 id="Classification_of_finite_simple_groups">Classification of finite simple groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=13" title="Edit section: Classification of finite simple groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> is a theorem stating that every <a href="/wiki/List_of_finite_simple_groups" title="List of finite simple groups">finite simple group</a> belongs to one of the following families: </p> <ul><li>A <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> with prime order;</li> <li>An <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> of degree at least 5;</li> <li>A <a href="/wiki/Group_of_Lie_type" title="Group of Lie type">simple group of Lie type</a>;</li> <li>One of the 26 <a href="/wiki/Sporadic_group" title="Sporadic group">sporadic simple groups</a>;</li> <li>The <a href="/wiki/Tits_group" title="Tits group">Tits group</a> (sometimes considered as a 27th sporadic group).</li></ul> <p>The finite simple groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the <a href="/wiki/Prime_number" title="Prime number">prime numbers</a> are the basic building blocks of the <a href="/wiki/Natural_number" title="Natural number">natural numbers</a>. The <a href="/wiki/Jordan%E2%80%93H%C3%B6lder_theorem" class="mw-redirect" title="Jordan–Hölder theorem">Jordan–Hölder theorem</a> is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a> is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same <a href="/wiki/Composition_series" title="Composition series">composition series</a> or, put in another way, the <a href="/wiki/Group_extension#Extension_problem" title="Group extension">extension problem</a> does not have a unique solution. </p><p>The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. <a href="/wiki/Daniel_Gorenstein" title="Daniel Gorenstein">Gorenstein</a> (d.1992), <a href="/wiki/Richard_Lyons_(mathematician)" title="Richard Lyons (mathematician)">Lyons</a>, and <a href="/wiki/Ronald_Solomon" title="Ronald Solomon">Solomon</a> are gradually publishing a simplified and revised version of the proof. </p> <div class="mw-heading mw-heading2"><h2 id="Number_of_groups_of_a_given_order">Number of groups of a given order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=14" title="Edit section: Number of groups of a given order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a positive integer <i>n</i>, it is not at all a routine matter to determine how many <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> types of groups of <a href="/wiki/Group_order" class="mw-redirect" title="Group order">order</a> <i>n</i> there are. Every group of <a href="/wiki/Prime_number" title="Prime number">prime</a> order is <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a>, because <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a> implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. If <i>n</i> is the square of a prime, then there are exactly two possible isomorphism types of group of order <i>n</i>, both of which are abelian. If <i>n</i> is a higher power of a prime, then results of <a href="/wiki/Graham_Higman" title="Graham Higman">Graham Higman</a> and <a href="/wiki/Charles_Sims_(mathematician)" title="Charles Sims (mathematician)">Charles Sims</a> give asymptotically correct estimates for the number of isomorphism types of groups of order <i>n</i>, and the number grows very rapidly as the power increases. </p><p>Depending on the prime factorization of <i>n</i>, some restrictions may be placed on the structure of groups of order <i>n</i>, as a consequence, for example, of results such as the <a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a>. For example, every group of order <i>pq</i> is cyclic when <span class="nowrap"><i>q</i> < <i>p</i></span> are primes with <span class="nowrap"><i>p</i> − 1</span> not divisible by <i>q</i>. For a necessary and sufficient condition, see <a href="/wiki/Cyclic_number_(group_theory)" title="Cyclic number (group theory)">cyclic number</a>. </p><p>If <i>n</i> is <a href="/wiki/Square-free_integer" title="Square-free integer">squarefree</a>, then any group of order <i>n</i> is solvable. <a href="/wiki/Burnside%27s_theorem" title="Burnside's theorem">Burnside's theorem</a>, proved using <a href="/wiki/Character_theory" title="Character theory">group characters</a>, states that every group of order <i>n</i> is solvable when <i>n</i> is divisible by fewer than three distinct primes, i.e. if <span class="nowrap"><i>n</i> = <i>p</i><sup><i>a</i></sup><i>q</i><sup><i>b</i></sup></span>, where <i>p</i> and <i>q</i> are prime numbers, and <i>a</i> and <i>b</i> are non-negative integers. By the <a href="/wiki/Feit%E2%80%93Thompson_theorem" title="Feit–Thompson theorem">Feit–Thompson theorem</a>, which has a long and complicated proof, every group of order <i>n</i> is solvable when <i>n</i> is odd. </p><p>For every positive integer <i>n</i>, most groups of order <i>n</i> are <a href="/wiki/Solvable_group" title="Solvable group">solvable</a>. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 12 solvable groups of order 60) but the proof of this for all orders uses the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>. For any positive integer <i>n</i> there are at most two simple groups of order <i>n</i>, and there are infinitely many positive integers <i>n</i> for which there are two non-isomorphic simple groups of order <i>n</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Table_of_distinct_groups_of_order_n">Table of distinct groups of order <i>n</i></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=15" title="Edit section: Table of distinct groups of order n"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="//oeis.org/A000001" class="extiw" title="oeis:A000001">oeis:A000001</a>, <a href="//oeis.org/A000688" class="extiw" title="oeis:A000688">oeis:A000688</a>, and <a href="//oeis.org/A060689" class="extiw" title="oeis:A060689">oeis:A060689</a></div> <table class="wikitable"> <tbody><tr> <th>Order <i>n</i> </th> <th># Groups<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </th> <th>Abelian </th> <th>Non-Abelian </th></tr> <tr> <th>0 </th> <td>0 </td> <td>0 </td> <td>0 </td></tr> <tr> <th>1 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>2 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>3 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>4 </th> <td>2 </td> <td>2 </td> <td>0 </td></tr> <tr> <th>5 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>6 </th> <td>2 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>7 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>8 </th> <td>5 </td> <td>3 </td> <td>2 </td></tr> <tr> <th>9 </th> <td>2 </td> <td>2 </td> <td>0 </td></tr> <tr> <th>10 </th> <td>2 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>11 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>12 </th> <td>5 </td> <td>2 </td> <td>3 </td></tr> <tr> <th>13 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>14 </th> <td>2 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>15 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>16 </th> <td>14 </td> <td>5 </td> <td>9 </td></tr> <tr> <th>17 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>18 </th> <td>5 </td> <td>2 </td> <td>3 </td></tr> <tr> <th>19 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>20 </th> <td>5 </td> <td>2 </td> <td>3 </td></tr> <tr> <th>21 </th> <td>2 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>22 </th> <td>2 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>23 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>24 </th> <td>15 </td> <td>3 </td> <td>12 </td></tr> <tr> <th>25 </th> <td>2 </td> <td>2 </td> <td>0 </td></tr> <tr> <th>26 </th> <td>2 </td> <td>1 </td> <td>1 </td></tr> <tr> <th>27 </th> <td>5 </td> <td>3 </td> <td>2 </td></tr> <tr> <th>28 </th> <td>4 </td> <td>2 </td> <td>2 </td></tr> <tr> <th>29 </th> <td>1 </td> <td>1 </td> <td>0 </td></tr> <tr> <th>30 </th> <td>4 </td> <td>1 </td> <td>3 </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 30em;"> <ul><li><a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></li> <li><a href="/wiki/Association_scheme" title="Association scheme">Association scheme</a></li> <li><a href="/wiki/List_of_finite_simple_groups" title="List of finite simple groups">List of finite simple groups</a></li> <li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem (group theory)</a></li> <li><a href="/wiki/P-group" title="P-group">P-group</a></li> <li><a href="/wiki/List_of_small_groups" title="List of small groups">List of small groups</a></li> <li><a href="/wiki/Representation_theory_of_finite_groups" title="Representation theory of finite groups">Representation theory of finite groups</a></li> <li><a href="/wiki/Modular_representation_theory" title="Modular representation theory">Modular representation theory</a></li> <li><a href="/wiki/Monstrous_moonshine" title="Monstrous moonshine">Monstrous moonshine</a></li> <li><a href="/wiki/Profinite_group" title="Profinite group">Profinite group</a></li> <li><a href="/wiki/Finite_ring" title="Finite ring">Finite ring</a></li> <li><a href="/wiki/Commuting_probability" title="Commuting probability">Commuting probability</a></li> <li><a href="/wiki/Finite_State_Machine" class="mw-redirect" title="Finite State Machine">Finite State Machine</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">Infinite group</a></li></ul></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=Finite_group&action=edit&section=17" title="Edit section: References"><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 reflist-columns references-column-width" style="column-width: 30em;"> <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"><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="CITEREFAschbacher2004" class="citation news cs1"><a href="/wiki/Michael_Aschbacher" title="Michael Aschbacher">Aschbacher, Michael</a> (2004). <a rel="nofollow" class="external text" href="https://www.ams.org/notices/200407/fea-aschbacher.pdf">"The Status of the Classification of the Finite Simple Groups"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Notices_of_the_American_Mathematical_Society" title="Notices of the American Mathematical Society">Notices of the American Mathematical Society</a></i>. Vol. 51, no. 7. pp. 736–740.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+of+the+American+Mathematical+Society&rft.atitle=The+Status+of+the+Classification+of+the+Finite+Simple+Groups&rft.volume=51&rft.issue=7&rft.pages=736-740&rft.date=2004&rft.aulast=Aschbacher&rft.aufirst=Michael&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F200407%2Ffea-aschbacher.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+group" class="Z3988"></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"><a href="/wiki/Daniel_Gorenstein" title="Daniel Gorenstein">Daniel Gorenstein</a> (1985), "The Enormous Theorem", <i>Scientific American</i>, December 1, 1985, vol. 253, no. 6, pp. 104–115.</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"><a rel="nofollow" class="external text" href="https://chem.libretexts.org/Core/Physical_and_Theoretical_Chemistry/Group_Theory/Group_Theory_and_its_Application_to_Chemistry">Group Theory and its Application to Chemistry</a> The Chemistry LibreTexts library</span> </li> <li id="cite_note-Jacobson-def-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Jacobson-def_4-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson 2009</a>, p. 31</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="#CITEREFJacobson2009">Jacobson 2009</a>, p. 41</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"><a href="#CITEREFJacobson2009">Jacobson 2009</a>, p. 38</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson 2009</a>, p. 72, ex. 1</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHumphreys1996" class="citation book cs1">Humphreys, John F. (1996). <i>A Course in Group Theory</i>. Oxford University Press. pp. 238–242. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0198534590" title="Special:BookSources/0198534590"><bdi>0198534590</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0843.20001">0843.20001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Group+Theory&rft.pages=238-242&rft.pub=Oxford+University+Press&rft.date=1996&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0843.20001%23id-name%3DZbl&rft.isbn=0198534590&rft.aulast=Humphreys&rft.aufirst=John+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+group" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=18" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson2009" class="citation book cs1">Jacobson, Nathan (2009). <i>Basic Algebra I</i> (2nd ed.). <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47189-1" title="Special:BookSources/978-0-486-47189-1"><bdi>978-0-486-47189-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Algebra+I&rft.edition=2nd&rft.pub=Dover+Publications&rft.date=2009&rft.isbn=978-0-486-47189-1&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFinite+group" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Finite_group&action=edit&section=19" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A000001">sequence A000001 (Number of groups of order n)</a></li> <li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A000688">sequence A000688 (Number of Abelian groups of order <i>n</i>)</a></li> <li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A060689">sequence A060689 (Number of non-Abelian groups of order n)</a></li> <li>Small groups on <a rel="nofollow" class="external text" href="http://groupnames.org">GroupNames</a></li> <li>A <a rel="nofollow" class="external text" href="http://www.bluetulip.org/programs/finitegroups.html">classifier</a> for groups of small order</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output 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