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class="vector-toc-numb">2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_properties_and_special_elements" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Group_properties_and_special_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Group properties and special elements</span> </div> </a> <button aria-controls="toc-Group_properties_and_special_elements-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 Group properties and special elements subsection</span> </button> <ul id="toc-Group_properties_and_special_elements-sublist" class="vector-toc-list"> <li id="toc-Multiplication" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplication"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Multiplication</span> </div> </a> <ul id="toc-Multiplication-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Verification_of_group_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Verification_of_group_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Verification of group axioms</span> </div> </a> <ul id="toc-Verification_of_group_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transpositions,_sign,_and_the_alternating_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transpositions,_sign,_and_the_alternating_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Transpositions, sign, and the alternating group</span> </div> </a> <ul id="toc-Transpositions,_sign,_and_the_alternating_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cycles" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cycles"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Cycles</span> </div> </a> <ul id="toc-Cycles-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_elements" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_elements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Special elements</span> </div> </a> <ul id="toc-Special_elements-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conjugacy_classes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conjugacy_classes"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conjugacy classes</span> </div> </a> <ul id="toc-Conjugacy_classes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Low_degree_groups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Low_degree_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Low degree groups</span> </div> </a> <button aria-controls="toc-Low_degree_groups-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 Low degree groups subsection</span> </button> <ul id="toc-Low_degree_groups-sublist" class="vector-toc-list"> <li id="toc-Maps_between_symmetric_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maps_between_symmetric_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Maps between symmetric groups</span> </div> </a> <ul id="toc-Maps_between_symmetric_groups-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relation_with_alternating_group" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_with_alternating_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Relation with alternating group</span> </div> </a> <ul id="toc-Relation_with_alternating_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generators_and_relations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generators_and_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generators and relations</span> </div> </a> <ul id="toc-Generators_and_relations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgroup_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Subgroup_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Subgroup structure</span> </div> </a> <button aria-controls="toc-Subgroup_structure-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 Subgroup structure subsection</span> </button> <ul id="toc-Subgroup_structure-sublist" class="vector-toc-list"> <li id="toc-Normal_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Normal subgroups</span> </div> </a> <ul id="toc-Normal_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Maximal_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Maximal_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Maximal subgroups</span> </div> </a> <ul id="toc-Maximal_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sylow_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sylow_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.3</span> <span>Sylow subgroups</span> </div> </a> <ul id="toc-Sylow_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transitive_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transitive_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.4</span> <span>Transitive subgroups</span> </div> </a> <ul id="toc-Transitive_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Young_subgroups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Young_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.5</span> <span>Young subgroups</span> </div> </a> <ul id="toc-Young_subgroups-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">8.6</span> <span>Cayley's theorem</span> </div> </a> <ul id="toc-Cayley's_theorem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cyclic_subgroups" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Cyclic_subgroups"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Cyclic subgroups</span> </div> </a> <ul id="toc-Cyclic_subgroups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Automorphism_group" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Automorphism_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Automorphism group</span> </div> </a> <ul id="toc-Automorphism_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homology" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Homology"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Homology</span> </div> </a> <ul id="toc-Homology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Representation_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Representation theory</span> </div> </a> <ul id="toc-Representation_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</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"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">16</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 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" 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Available in 26 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-26" 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">26 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D9%85%D8%AA%D9%86%D8%A7%D8%B8%D8%B1%D8%A9" title="زمرة متناظرة – Arabic" lang="ar" hreflang="ar" data-title="زمرة متناظرة" 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/Grup_sim%C3%A8tric" title="Grup simètric – Catalan" lang="ca" hreflang="ca" data-title="Grup simètric" 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/Symetrick%C3%A1_grupa" title="Symetrická grupa – Czech" lang="cs" hreflang="cs" data-title="Symetrická grupa" 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/Symmetrische_Gruppe" title="Symmetrische Gruppe – German" lang="de" hreflang="de" data-title="Symmetrische 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_sim%C3%A9trico" title="Grupo simétrico – Spanish" lang="es" hreflang="es" data-title="Grupo simétrico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Simetria_grupo" title="Simetria grupo – Esperanto" lang="eo" hreflang="eo" data-title="Simetria grupo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde_simetriko" title="Talde simetriko – Basque" lang="eu" hreflang="eu" data-title="Talde simetriko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</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%82%D8%A7%D8%B1%D9%86" title="گروه متقارن – Persian" lang="fa" hreflang="fa" data-title="گروه متقارن" 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_sym%C3%A9trique" title="Groupe symétrique – French" lang="fr" hreflang="fr" data-title="Groupe symétrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Grupo_sim%C3%A9trico" title="Grupo simétrico – Galician" lang="gl" hreflang="gl" data-title="Grupo simétrico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EC%B9%AD%EA%B5%B0_(%EA%B5%B0%EB%A1%A0)" title="대칭군 (군론) – Korean" lang="ko" hreflang="ko" data-title="대칭군 (군론)" 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_simetrik" title="Grup simetrik – Indonesian" lang="id" hreflang="id" data-title="Grup simetrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_symmetric" title="Gruppo symmetric – Interlingua" lang="ia" hreflang="ia" data-title="Gruppo symmetric" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_simmetrico" title="Gruppo simmetrico – Italian" lang="it" hreflang="it" data-title="Gruppo simmetrico" 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%94%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%94%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="החבורה הסימטרית – Hebrew" lang="he" hreflang="he" data-title="החבורה הסימטרית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B8%E0%B4%AE%E0%B4%AE%E0%B4%BF%E0%B4%A4%E0%B5%80%E0%B4%AF%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D" title="സമമിതീയഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="സമമിതീയഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Symmetrische_groep" title="Symmetrische groep – Dutch" lang="nl" hreflang="nl" data-title="Symmetrische 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/%E5%AF%BE%E7%A7%B0%E7%BE%A4" title="対称群 – Japanese" lang="ja" hreflang="ja" data-title="対称群" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Симметрическая группа – Russian" lang="ru" hreflang="ru" data-title="Симметрическая группа" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Simetri%C4%8Dna_grupa" title="Simetrična grupa – Slovenian" lang="sl" hreflang="sl" data-title="Simetrična grupa" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Симетрична група – Serbian" lang="sr" hreflang="sr" data-title="Симетрична 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Symmetrinen_ryhm%C3%A4" title="Symmetrinen ryhmä – Finnish" lang="fi" hreflang="fi" data-title="Symmetrinen ryhmä" 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/Symmetrisk_grupp" title="Symmetrisk grupp – Swedish" lang="sv" hreflang="sv" data-title="Symmetrisk grupp" data-language-autonym="Svenska" data-language-local-name="Swedish" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of group in abstract algebra</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">Not to be confused with <a href="/wiki/Symmetry_group" title="Symmetry group">Symmetry 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 S<sub>4</sub> using the generators (red) a right <a href="/wiki/Circular_shift" title="Circular shift">circular shift</a> of all four set elements, and (blue) a left circular shift of the first three set elements.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Symmetric_group_3;_Cayley_table;_matrices.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Symmetric_group_3%3B_Cayley_table%3B_matrices.svg/320px-Symmetric_group_3%3B_Cayley_table%3B_matrices.svg.png" decoding="async" width="320" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Symmetric_group_3%3B_Cayley_table%3B_matrices.svg/480px-Symmetric_group_3%3B_Cayley_table%3B_matrices.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Symmetric_group_3%3B_Cayley_table%3B_matrices.svg/640px-Symmetric_group_3%3B_Cayley_table%3B_matrices.svg.png 2x" data-file-width="2197" data-file-height="2197" /></a><figcaption><a href="/wiki/Cayley_table" title="Cayley table">Cayley table</a>, with <a href="/wiki/Table_(information)#:~:text=header" title="Table (information)">header</a> omitted, of the symmetric group S<sub>3</sub>. The elements are represented as <a href="/wiki/Permutation#Matrix_representation" title="Permutation">matrices</a>. To the left of the matrices, are their <a href="/wiki/Permutation#Two-line_notation" title="Permutation">two-line form</a>. The black arrows indicate disjoint cycles and correspond to <a href="/wiki/Permutation#Cycle_notation" title="Permutation">cycle notation</a>. Green circle is an odd permutation, white is an even permutation and black is the identity. <br /><br />These are the positions of the six matrices<br /><span typeof="mw:File"><a href="/wiki/File:Symmetric_group_3;_Cayley_table;_positions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Symmetric_group_3%3B_Cayley_table%3B_positions.svg/310px-Symmetric_group_3%3B_Cayley_table%3B_positions.svg.png" decoding="async" width="310" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Symmetric_group_3%3B_Cayley_table%3B_positions.svg/465px-Symmetric_group_3%3B_Cayley_table%3B_positions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Symmetric_group_3%3B_Cayley_table%3B_positions.svg/620px-Symmetric_group_3%3B_Cayley_table%3B_positions.svg.png 2x" data-file-width="768" data-file-height="144" /></a></span><br />Some matrices are not arranged symmetrically to the main diagonal – thus the symmetric group is not abelian.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><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 class="mw-selflink selflink">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>, the <b>symmetric group</b> defined over any <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> is the <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> whose <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> are all the <a href="/wiki/Bijection" title="Bijection">bijections</a> from the set to itself, 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 of functions</a>. In particular, the finite symmetric 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 \mathrm {S} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f98267cda84a9c8abfee60f7184af3cb1aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{n}}"></span> defined over a <a href="/wiki/Finite_set" title="Finite set">finite set</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> symbols consists of the <a href="/wiki/Permutation" title="Permutation">permutations</a> that can be performed on the <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> symbols.<sup id="cite_ref-Jacobson-def_1-0" class="reference"><a href="#cite_note-Jacobson-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Since there are <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 n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> <a href="/wiki/Factorial" title="Factorial">factorial</a>) such permutation operations, the <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</a> (number of elements) of the symmetric 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 \mathrm {S} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f98267cda84a9c8abfee60f7184af3cb1aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{n}}"></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 n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span>. </p><p>Although symmetric groups can be defined on <a href="/wiki/Infinite_set" title="Infinite set">infinite sets</a>, this article focuses on the finite symmetric groups: their applications, their elements, their <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy classes</a>, a <a href="/wiki/Finitely_presented_group" class="mw-redirect" title="Finitely presented group">finite presentation</a>, their <a href="/wiki/Subgroup" title="Subgroup">subgroups</a>, their <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism groups</a>, and their <a href="/wiki/Group_representation" title="Group representation">representation</a> theory. For the remainder of this article, "symmetric group" will mean a symmetric group on a finite set. </p><p>The symmetric group is important to diverse areas of mathematics such as <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, <a href="/wiki/Invariant_theory" title="Invariant theory">invariant theory</a>, the <a href="/wiki/Representation_theory_of_Lie_groups" class="mw-redirect" title="Representation theory of Lie groups">representation theory of Lie groups</a>, and <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>. <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a> states that every 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> is <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic</a> to a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the symmetric group on (the <a href="/wiki/Underlying_set" class="mw-redirect" title="Underlying set">underlying set</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 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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_first_properties">Definition and first properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=1" title="Edit section: Definition and first properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symmetric group on a finite set <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is the group whose elements are all bijective functions from <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> and whose group operation is that of <a href="/wiki/Function_composition" title="Function composition">function composition</a>.<sup id="cite_ref-Jacobson-def_1-1" class="reference"><a href="#cite_note-Jacobson-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For finite sets, "permutations" and "bijective functions" refer to the same operation, namely rearrangement. The symmetric group of <b>degree</b> <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the symmetric group on the set <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=\{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{1,2,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f30567ea1096cad88c3a13c30642eeb6aa08d7c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.335ex; height:2.843ex;" alt="{\displaystyle X=\{1,2,\ldots ,n\}}"></span>. </p><p>The symmetric group on a set <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is denoted in various ways, including <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 \mathrm {S} _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021c33a3f47a27709a73bd89d887bc8b6f49d3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.925ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{X}}"></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 {\mathfrak {S}}_{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}_{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e578e741e7bcf81385426c3d9ef1ed670a9bdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.559ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {S}}_{X}}"></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 \Sigma _{X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14aa7e50427b1aab436c19143994e835d596e458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.31ex; height:2.509ex;" alt="{\displaystyle \Sigma _{X}}"></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 X!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbadb5c2252c9448fef9b280fae7a88ab82dd17e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X!}"></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 \operatorname {Sym} (X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sym</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (X)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64f4a9bd46fe22dca3aeaa37d032281966a79a67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.245ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (X)}"></span>.<sup id="cite_ref-Jacobson-def_1-2" class="reference"><a href="#cite_note-Jacobson-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is the set <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 \{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebfec86b3f22a18f086275390917d5aaa2d8c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}}"></span> then the name may be abbreviated 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 \mathrm {S} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f98267cda84a9c8abfee60f7184af3cb1aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{n}}"></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 {\mathfrak {S}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {S}}_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be1a166cbba7777116f34a11599cdb6de0620691" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.145ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {S}}_{n}}"></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 \Sigma _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fe788877efb85808d238e1b1f47c5c290f2384b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.897ex; height:2.509ex;" alt="{\displaystyle \Sigma _{n}}"></span>, or <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 \operatorname {Sym} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sym</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b53f7a393ab1e376574e61af57e6cd8591b0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.66ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (n)}"></span>.<sup id="cite_ref-Jacobson-def_1-3" class="reference"><a href="#cite_note-Jacobson-def-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Symmetric groups on infinite sets behave quite differently from symmetric groups on finite sets, and are discussed in (<a href="#CITEREFScott1987">Scott 1987</a>, Ch. 11), (<a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, Ch. 8), and (<a href="#CITEREFCameron1999">Cameron 1999</a>). </p><p>The symmetric group on a set 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> elements has <a href="/wiki/Order_(group_theory)" title="Order (group theory)">order</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 n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"></span> (the <a href="/wiki/Factorial" title="Factorial">factorial</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>).<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> It is <a href="/wiki/Abelian_group" title="Abelian group">abelian</a> if and only 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is less than or equal to 2.<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> For <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 n=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26819344e55f5e671c76c07c18eb4291fcec85ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=0}"></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 n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span> (the <a href="/wiki/Empty_set" title="Empty set">empty set</a> and the <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a>), the symmetric groups are <a href="/wiki/Trivial_group" title="Trivial group">trivial</a> (they have order <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 0!=1!=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> <mo>!</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0!=1!=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df097d5656bdb793a700f4c9bee417f5a009fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.978ex; height:2.176ex;" alt="{\displaystyle 0!=1!=1}"></span>). The group S<sub><i>n</i></sub> is <a href="/wiki/Solvable_group" title="Solvable group">solvable</a> if and only 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 n\leq 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≤</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\leq 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/602ee104eb13147da2d210d20c294982fcf93713" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\leq 4}"></span>. This is an essential part of the proof of the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> that shows that for every <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 n>4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6b13dc8b113121cdaf76a723a61aa4f8be1468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>4}"></span> there are <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> of degree <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> which are not solvable by radicals, that is, the solutions cannot be expressed by performing a finite number of operations of addition, subtraction, multiplication, division and root extraction on the polynomial's coefficients. </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=Symmetric_group&action=edit&section=2" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symmetric group on a set of size <i>n</i> is the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of the general <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> of degree <i>n</i> and plays an important role in <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>. In <a href="/wiki/Invariant_theory" title="Invariant theory">invariant theory</a>, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called <a href="/wiki/Symmetric_function" title="Symmetric function">symmetric functions</a>. In the <a href="/wiki/Representation_theory_of_Lie_groups" class="mw-redirect" title="Representation theory of Lie groups">representation theory of Lie groups</a>, the <a href="/wiki/Representation_theory_of_the_symmetric_group" title="Representation theory of the symmetric group">representation theory of the symmetric group</a> plays a fundamental role through the ideas of <a href="/wiki/Schur_functor" title="Schur functor">Schur functors</a>. </p><p>In the theory of <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a>, the symmetric group is the Coxeter group of type A<sub><i>n</i></sub> and occurs as the <a href="/wiki/Weyl_group" title="Weyl group">Weyl group</a> of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>. In <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, the symmetric groups, their elements (<a href="/wiki/Permutation" title="Permutation">permutations</a>), and their <a href="/wiki/Group_representation" title="Group representation">representations</a> provide a rich source of problems involving <a href="/wiki/Young_tableaux" class="mw-redirect" title="Young tableaux">Young tableaux</a>, <a href="/wiki/Plactic_monoid" title="Plactic monoid">plactic monoids</a>, and the <a href="/wiki/Bruhat_order" title="Bruhat order">Bruhat order</a>. <a href="/wiki/Subgroup" title="Subgroup">Subgroups</a> of symmetric groups are called <a href="/wiki/Permutation_group" title="Permutation group">permutation groups</a> and are widely studied because of their importance in understanding <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group actions</a>, <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous spaces</a>, and <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism groups</a> of <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, such as the <a href="/wiki/Higman%E2%80%93Sims_group" title="Higman–Sims group">Higman–Sims group</a> and the <a href="/wiki/Higman%E2%80%93Sims_graph" title="Higman–Sims graph">Higman–Sims graph</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Group_properties_and_special_elements">Group properties and special elements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=3" title="Edit section: Group properties and special elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The elements of the symmetric group on a set <i>X</i> are the <a href="/wiki/Permutation" title="Permutation">permutations</a> of <i>X</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Multiplication">Multiplication</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=4" title="Edit section: Multiplication"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The group operation in a symmetric group is function composition, denoted by the symbol ∘ or by simple juxtaposition. The composition <span class="texhtml"><i>f</i> ∘ <i>g</i></span> of permutations <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span>, pronounced "<span class="texhtml mvar" style="font-style:italic;">f</span> of <span class="texhtml mvar" style="font-style:italic;">g</span>", maps any element <span class="texhtml mvar" style="font-style:italic;">x</span> of <span class="texhtml mvar" style="font-style:italic;">X</span> to <span class="texhtml"><i>f</i>(<i>g</i>(<i>x</i>))</span>. Concretely, let (see <a href="/wiki/Permutation" title="Permutation">permutation</a> for an explanation of notation): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=(1~3)(2)(4~5)={\begin{pmatrix}1&2&3&4&5\\3&2&1&5&4\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mtext> </mtext> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=(1~3)(2)(4~5)={\begin{pmatrix}1&2&3&4&5\\3&2&1&5&4\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb6fea51c7a5cdcf2a3fa946b9b30152280a3891" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.799ex; height:6.176ex;" alt="{\displaystyle f=(1~3)(2)(4~5)={\begin{pmatrix}1&2&3&4&5\\3&2&1&5&4\end{pmatrix}},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=(1~2~5)(3~4)={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mtext> </mtext> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=(1~2~5)(3~4)={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0719fd2e9a58cc8b4fbe7411ecad22ab2e6c99f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.408ex; height:6.176ex;" alt="{\displaystyle g=(1~2~5)(3~4)={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}.}"></span> Applying <span class="texhtml mvar" style="font-style:italic;">f</span> after <span class="texhtml mvar" style="font-style:italic;">g</span> maps 1 first to 2 and then 2 to itself; 2 to 5 and then to 4; 3 to 4 and then to 5, and so on. So, composing <span class="texhtml mvar" style="font-style:italic;">f</span> and <span class="texhtml mvar" style="font-style:italic;">g</span> gives <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg=f\circ g=(1\ 2\ 4)(3\ 5)={\begin{pmatrix}1&2&3&4&5\\2&4&5&1&3\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>∘</mo> <mi>g</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mtext> </mtext> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle fg=f\circ g=(1\ 2\ 4)(3\ 5)={\begin{pmatrix}1&2&3&4&5\\2&4&5&1&3\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72f11c017b2f6f43f06c0835ef732aa608d637e1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.374ex; height:6.176ex;" alt="{\displaystyle fg=f\circ g=(1\ 2\ 4)(3\ 5)={\begin{pmatrix}1&2&3&4&5\\2&4&5&1&3\end{pmatrix}}.}"></span> </p><p>A <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cycle</a> of length <span class="texhtml"><i>L</i> = <i>k</i> · <i>m</i></span>, taken to the <span class="texhtml mvar" style="font-style:italic;">k</span>th power, will decompose into <span class="texhtml mvar" style="font-style:italic;">k</span> cycles of length <span class="texhtml mvar" style="font-style:italic;">m</span>: For example, (<span class="texhtml"><i>k</i> = 2</span>, <span class="texhtml"><i>m</i> = 3</span>), <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1~2~3~4~5~6)^{2}=(1~3~5)(2~4~6).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mn>3</mn> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> <mn>5</mn> <mtext> </mtext> <mn>6</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>3</mn> <mtext> </mtext> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> <mn>6</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1~2~3~4~5~6)^{2}=(1~3~5)(2~4~6).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fde89c235dcc3fa48e23ff102a4577dd87181d2b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.403ex; height:3.176ex;" alt="{\displaystyle (1~2~3~4~5~6)^{2}=(1~3~5)(2~4~6).}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Verification_of_group_axioms">Verification of group axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=5" title="Edit section: Verification of group axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To check that the symmetric group on a set <i>X</i> is indeed a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, it is necessary to verify the group axioms of closure, associativity, identity, and inverses.<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> </p> <ol><li>The operation of function composition is closed in the set of permutations of the given set <i>X</i>.</li> <li>Function composition is always associative.</li> <li>The trivial bijection that assigns each element of <i>X</i> to itself serves as an identity for the group.</li> <li>Every bijection has an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a> that undoes its action, and thus each element of a symmetric group does have an inverse which is a permutation too.</li></ol> <div class="mw-heading mw-heading3"><h3 id="Transpositions,_sign,_and_the_alternating_group"><span id="Transpositions.2C_sign.2C_and_the_alternating_group"></span>Transpositions, sign, and the alternating group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=6" title="Edit section: Transpositions, sign, and the alternating group"><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/Transposition_(mathematics)" class="mw-redirect" title="Transposition (mathematics)">Transposition (mathematics)</a></div> <p>A <b>transposition</b> is a permutation which exchanges two elements and keeps all others fixed; for example (1 3) is a transposition. Every permutation can be written as a product of transpositions; for instance, the permutation <i>g</i> from above can be written as <i>g</i> = (1 2)(2 5)(3 4). Since <i>g</i> can be written as a product of an odd number of transpositions, it is then called an <a href="/wiki/Even_and_odd_permutations" class="mw-redirect" title="Even and odd permutations">odd permutation</a>, whereas <i>f</i> is an even permutation. </p><p>The representation of a permutation as a product of transpositions is not unique; however, the number of transpositions needed to represent a given permutation is either always even or always odd. There are several short proofs of the invariance of this parity of a permutation. </p><p>The product of two even permutations is even, the product of two odd permutations is even, and the product of one of each is odd. Thus we can define the <b>sign</b> of a permutation: </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 \operatorname {sgn} f={\begin{cases}+1,&{\text{if }}f{\mbox{ is even}}\\-1,&{\text{if }}f{\text{ is odd}}.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> is even</mtext> </mstyle> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd</mtext> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} f={\begin{cases}+1,&{\text{if }}f{\mbox{ is even}}\\-1,&{\text{if }}f{\text{ is odd}}.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cabfcb749228a5d4ba050be293d9327bf77c4907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.097ex; height:6.176ex;" alt="{\displaystyle \operatorname {sgn} f={\begin{cases}+1,&{\text{if }}f{\mbox{ is even}}\\-1,&{\text{if }}f{\text{ is odd}}.\end{cases}}}"></span></dd></dl> <p>With this definition, </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 \operatorname {sgn} \colon \mathrm {S} _{n}\rightarrow \{+1,-1\}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>:</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mo fence="false" stretchy="false">{</mo> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} \colon \mathrm {S} _{n}\rightarrow \{+1,-1\}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/939505ed04dba147734fdc8eb0009501b06a97b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.411ex; height:2.843ex;" alt="{\displaystyle \operatorname {sgn} \colon \mathrm {S} _{n}\rightarrow \{+1,-1\}\ }"></span></dd></dl> <p>is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a> ({+1, −1} is a group under multiplication, where +1 is e, the <a href="/wiki/Neutral_element" class="mw-redirect" title="Neutral element">neutral element</a>). The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of this homomorphism, that is, the set of all even permutations, is called the <b><a href="/wiki/Alternating_group" title="Alternating group">alternating group</a></b> A<sub><i>n</i></sub>. It is a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> of S<sub><i>n</i></sub>, and for <span class="nowrap"><i>n</i> ≥ 2</span> it has <span class="nowrap"><i>n</i>!/2</span> elements. The group S<sub><i>n</i></sub> is the <a href="/wiki/Semidirect_product" title="Semidirect product">semidirect product</a> of A<sub><i>n</i></sub> and any subgroup generated by a single transposition. </p><p>Furthermore, every permutation can be written as a product of <i><a href="/wiki/Adjacent_transposition" class="mw-redirect" title="Adjacent transposition">adjacent transpositions</a></i>, that is, transpositions of the form <span class="nowrap">(<i>a</i> <i>a</i>+1)</span>. For instance, the permutation <i>g</i> from above can also be written as <span class="nowrap"><i>g</i> = (4 5)(3 4)(4 5)(1 2)(2 3)(3 4)(4 5)</span>. The sorting algorithm <a href="/wiki/Bubble_sort" title="Bubble sort">bubble sort</a> is an application of this fact. The representation of a permutation as a product of adjacent transpositions is also not unique. </p> <div class="mw-heading mw-heading3"><h3 id="Cycles">Cycles</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=7" title="Edit section: Cycles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Cyclic_permutation" title="Cyclic permutation">cycle</a> of <i>length</i> <i>k</i> is a permutation <i>f</i> for which there exists an element <i>x</i> in {1, ..., <i>n</i>} such that <i>x</i>, <i>f</i>(<i>x</i>), <i>f</i><sup>2</sup>(<i>x</i>), ..., <i>f</i><sup><i>k</i></sup>(<i>x</i>) = <i>x</i> are the only elements moved by <i>f</i>; it conventionally is required that <span class="nowrap"><i>k</i> ≥ 2</span> since with <span class="nowrap"><i>k</i> = 1</span> the element <i>x</i> itself would not be moved either. The permutation <i>h</i> defined by </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 h={\begin{pmatrix}1&2&3&4&5\\4&2&1&3&5\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h={\begin{pmatrix}1&2&3&4&5\\4&2&1&3&5\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbc00db9826afa3aa6e50819832521445fb35e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.713ex; height:6.176ex;" alt="{\displaystyle h={\begin{pmatrix}1&2&3&4&5\\4&2&1&3&5\end{pmatrix}}}"></span></dd></dl> <p>is a cycle of length three, since <span class="nowrap"><i>h</i>(1) = 4</span>, <span class="nowrap"><i>h</i>(4) = 3</span> and <span class="nowrap"><i>h</i>(3) = 1</span>, leaving 2 and 5 untouched. We denote such a cycle by <span class="nowrap">(1 4 3)</span>, but it could equally well be written <span class="nowrap">(4 3 1)</span> or <span class="nowrap">(3 1 4)</span> by starting at a different point. The order of a cycle is equal to its length. Cycles of length two are transpositions. Two cycles are <i>disjoint</i> if they have disjoint subsets of elements. Disjoint cycles <a href="/wiki/Commutative_property" title="Commutative property">commute</a>: for example, in S<sub>6</sub> there is the equality <span class="nowrap">(4 1 3)(2 5 6) = (2 5 6)(4 1 3)</span>. Every element of S<sub><i>n</i></sub> can be written as a product of disjoint cycles; this representation is unique <a href="/wiki/Up_to" title="Up to">up to</a> the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point. </p><p>Cycles admit the following conjugation property with any permutation <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>, this property is often used to obtain its <a class="mw-selflink-fragment" href="#Generators_and_relations">generators and relations</a>. </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 \sigma {\begin{pmatrix}a&b&c&\ldots \end{pmatrix}}\sigma ^{-1}={\begin{pmatrix}\sigma (a)&\sigma (b)&\sigma (c)&\ldots \end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> <mtd> <mo>…</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>…</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma {\begin{pmatrix}a&b&c&\ldots \end{pmatrix}}\sigma ^{-1}={\begin{pmatrix}\sigma (a)&\sigma (b)&\sigma (c)&\ldots \end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242cc4ceb652db4a505559ab188161558c5577b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.48ex; height:3.176ex;" alt="{\displaystyle \sigma {\begin{pmatrix}a&b&c&\ldots \end{pmatrix}}\sigma ^{-1}={\begin{pmatrix}\sigma (a)&\sigma (b)&\sigma (c)&\ldots \end{pmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Special_elements">Special elements</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=8" title="Edit section: Special elements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Certain elements of the symmetric group of {1, 2, ..., <i>n</i>} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set). </p><p>The <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="order_reversing_permutation"></span><span class="vanchor-text">order reversing permutation</span></span></b> is the one given by: </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 {\begin{pmatrix}1&2&\cdots &n\\n&n-1&\cdots &1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&2&\cdots &n\\n&n-1&\cdots &1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ffd9bb78b0a72e854f21113a33a5ba2dcb92db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.697ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&2&\cdots &n\\n&n-1&\cdots &1\end{pmatrix}}.}"></span></dd></dl> <p>This is the unique maximal element with respect to the <a href="/wiki/Bruhat_order" title="Bruhat order">Bruhat order</a> and the <a href="/wiki/Longest_element_of_a_Coxeter_group" title="Longest element of a Coxeter group">longest element</a> in the symmetric group with respect to generating set consisting of the adjacent transpositions <span class="nowrap">(<i>i</i> <i>i</i>+1)</span>, <span class="nowrap">1 ≤ <i>i</i> ≤ <i>n</i> − 1</span>. </p><p>This is an involution, and consists 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 \lfloor n/2\rfloor }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lfloor n/2\rfloor }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86b5dc915aaa6792f2a7d3ed1c165c555256c7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.784ex; height:2.843ex;" alt="{\displaystyle \lfloor n/2\rfloor }"></span> (non-adjacent) transpositions </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 (1\,n)(2\,n-1)\cdots ,{\text{ or }}\sum _{k=1}^{n-1}k={\frac {n(n-1)}{2}}{\text{ adjacent transpositions: }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mi>n</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> or </mtext> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> adjacent transpositions: </mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1\,n)(2\,n-1)\cdots ,{\text{ or }}\sum _{k=1}^{n-1}k={\frac {n(n-1)}{2}}{\text{ adjacent transpositions: }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065b80a78d7003c553f106f7a92a02b744d0da0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:64.262ex; height:7.343ex;" alt="{\displaystyle (1\,n)(2\,n-1)\cdots ,{\text{ or }}\sum _{k=1}^{n-1}k={\frac {n(n-1)}{2}}{\text{ adjacent transpositions: }}}"></span> <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 (n\,n-1)(n-1\,n-2)\cdots (2\,1)(n-1\,n-2)(n-2\,n-3)\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mspace width="thinmathspace" /> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n\,n-1)(n-1\,n-2)\cdots (2\,1)(n-1\,n-2)(n-2\,n-3)\cdots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/601d1b494bfdad6c6646b241b8925e0d4bda6446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.126ex; height:2.843ex;" alt="{\displaystyle (n\,n-1)(n-1\,n-2)\cdots (2\,1)(n-1\,n-2)(n-2\,n-3)\cdots ,}"></span></dd></dl></dd></dl> <p>so it thus has sign: </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 \mathrm {sgn} (\rho _{n})=(-1)^{\lfloor n/2\rfloor }=(-1)^{n(n-1)/2}={\begin{cases}+1&n\equiv 0,1{\pmod {4}}\\-1&n\equiv 2,3{\pmod {4}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">n</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mi>n</mi> <mo>≡</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mi>n</mi> <mo>≡</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="0.444em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>4</mn> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {sgn} (\rho _{n})=(-1)^{\lfloor n/2\rfloor }=(-1)^{n(n-1)/2}={\begin{cases}+1&n\equiv 0,1{\pmod {4}}\\-1&n\equiv 2,3{\pmod {4}}\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f4efb3709fab4b220fc77a15e1c412465a8736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.203ex; height:6.176ex;" alt="{\displaystyle \mathrm {sgn} (\rho _{n})=(-1)^{\lfloor n/2\rfloor }=(-1)^{n(n-1)/2}={\begin{cases}+1&n\equiv 0,1{\pmod {4}}\\-1&n\equiv 2,3{\pmod {4}}\end{cases}}}"></span></dd></dl> <p>which is 4-periodic in <i>n</i>. </p><p>In S<sub>2<i>n</i></sub>, the <i><a href="/wiki/Faro_shuffle" title="Faro shuffle">perfect shuffle</a></i> is the permutation that splits the set into 2 piles and interleaves them. Its sign is also <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 (-1)^{\lfloor n/2\rfloor }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)^{\lfloor n/2\rfloor }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e4c89dbca10414e51c5d8c30a2f4b09f3b158e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.749ex; height:3.343ex;" alt="{\displaystyle (-1)^{\lfloor n/2\rfloor }.}"></span> </p><p>Note that the reverse on <i>n</i> elements and perfect shuffle on 2<i>n</i> elements have the same sign; these are important to the classification of <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a>, which are 8-periodic. </p> <div class="mw-heading mw-heading2"><h2 id="Conjugacy_classes">Conjugacy classes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=9" title="Edit section: Conjugacy classes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy classes</a> of S<sub><i>n</i></sub> correspond to the <a href="/wiki/Permutation#Cycle_type" title="Permutation">cycle types</a> of permutations; that is, two elements of S<sub><i>n</i></sub> are conjugate in S<sub><i>n</i></sub> if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S<sub>5</sub>, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of S<sub><i>n</i></sub> can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k={\begin{pmatrix}1&2&3&4&5\\1&4&3&2&5\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k={\begin{pmatrix}1&2&3&4&5\\1&4&3&2&5\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d4e564a00f76cd71176e9886de3b781008ccddc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.232ex; height:6.176ex;" alt="{\displaystyle k={\begin{pmatrix}1&2&3&4&5\\1&4&3&2&5\end{pmatrix}},}"></span> which can be written as the product of cycles as (2 4). This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2~4)\circ (1~2~3)(4~5)\circ (2~4)=(1~4~3)(2~5).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mtext> </mtext> <mn>4</mn> <mo stretchy="false">)</mo> <mo>∘</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>2</mn> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mtext> </mtext> <mn>5</mn> <mo stretchy="false">)</mo> <mo>∘</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mtext> </mtext> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mtext> </mtext> <mn>4</mn> <mtext> </mtext> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mtext> </mtext> <mn>5</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2~4)\circ (1~2~3)(4~5)\circ (2~4)=(1~4~3)(2~5).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c56d85d44be21ad2c8b2470f0238f85e8286d1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.91ex; height:2.843ex;" alt="{\displaystyle (2~4)\circ (1~2~3)(4~5)\circ (2~4)=(1~4~3)(2~5).}"></span> It is clear that such a permutation is not unique. </p><p>Conjugacy classes of S<sub><i>n</i></sub> correspond to <a href="/wiki/Integer_partition" title="Integer partition">integer partitions</a> of <i>n</i>: to the partition <span class="nowrap"><i>μ</i> = (<i>μ</i><sub>1</sub>, <i>μ</i><sub>2</sub>, ..., <i>μ</i><sub><i>k</i></sub>)</span> with <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="{\textstyle n=\sum _{i=1}^{k}\mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n=\sum _{i=1}^{k}\mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb6f9f1801a33542aa60af8361696b94d2e3cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.436ex; height:3.509ex;" alt="{\textstyle n=\sum _{i=1}^{k}\mu _{i}}"></span> and <span class="nowrap"><i>μ</i><sub>1</sub> ≥ <i>μ</i><sub>2</sub> ≥ ... ≥ <i>μ</i><sub><i>k</i></sub></span>, is associated the set <i>C</i><sub><i>μ</i></sub> of permutations with cycles of lengths <span class="nowrap"><i>μ</i><sub>1</sub>, <i>μ</i><sub>2</sub>, ..., <i>μ</i><sub><i>k</i></sub></span>. Then <i>C</i><sub><i>μ</i></sub> is a conjugacy class of S<sub><i>n</i></sub>, whose elements are said to be of cycle-type <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Low_degree_groups">Low degree groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=10" title="Edit section: Low degree 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">See also: <a href="/wiki/Representation_theory_of_the_symmetric_group#Special_cases" title="Representation theory of the symmetric group">Representation theory of the symmetric group § Special cases</a></div> <p>The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately. </p> <dl><dt>S<sub>0</sub> and S<sub>1</sub></dt> <dd>The symmetric groups on the <a href="/wiki/Empty_set" title="Empty set">empty set</a> and the <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton set</a> are trivial, which corresponds to <span class="texhtml">0! = 1! = 1</span>. In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of S<sub>0</sub>, its only member is the <a href="/wiki/Empty_function" class="mw-redirect" title="Empty function">empty function</a>.</dd></dl> <dl><dt>S<sub>2</sub></dt> <dd>This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> and is thus <a href="/wiki/Abelian_group" title="Abelian group">abelian</a>. In <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, this corresponds to the fact that the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a> gives a direct solution to the general <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomial</a> after extracting only a single root. In <a href="/wiki/Invariant_theory" title="Invariant theory">invariant theory</a>, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting <span class="texhtml"><i>f</i><sub>s</sub>(<i>x</i>, <i>y</i>) = <i>f</i>(<i>x</i>, <i>y</i>) + <i>f</i>(<i>y</i>, <i>x</i>)</span>, and <span class="texhtml"><i>f</i><sub>a</sub>(<i>x</i>, <i>y</i>) = <i>f</i>(<i>x</i>, <i>y</i>) − <i>f</i>(<i>y</i>, <i>x</i>)</span>, one gets that <span class="texhtml">2⋅<i>f</i> = <i>f</i><sub>s</sub> + <i>f</i><sub>a</sub></span>. This process is known as <a href="/wiki/Symmetrization" title="Symmetrization">symmetrization</a>.</dd></dl> <dl><dt>S<sub>3</sub></dt> <dd>S<sub>3</sub> is the first nonabelian symmetric group. This group is isomorphic to the <a href="/wiki/Dihedral_group_of_order_6" title="Dihedral group of order 6">dihedral group of order 6</a>, the group of reflection and rotation symmetries of an <a href="/wiki/Equilateral_triangle" title="Equilateral triangle">equilateral triangle</a>, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from S<sub>3</sub> to S<sub>2</sub> corresponds to the resolving quadratic for a <a href="/wiki/Cubic_polynomial" class="mw-redirect" title="Cubic polynomial">cubic polynomial</a>, as discovered by <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a>, while the A<sub>3</sub> kernel corresponds to the use of the <a href="/wiki/Discrete_Fourier_transform" title="Discrete Fourier transform">discrete Fourier transform</a> of order 3 in the solution, in the form of <a href="/wiki/Lagrange_resolvent" class="mw-redirect" title="Lagrange resolvent">Lagrange resolvents</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2009)">citation needed</span></a></i>]</sup></dd></dl> <dl><dt>S<sub>4</sub></dt> <dd>The group S<sub>4</sub> is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, <a href="/wiki/Rencontres_numbers" title="Rencontres numbers">9, 8 and 6</a> permutations, of the <a href="/wiki/Cube" title="Cube">cube</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> Beyond the group <a href="/wiki/Alternating_group" title="Alternating group">A<sub>4</sub></a>, S<sub>4</sub> has a <a href="/wiki/Klein_four-group" title="Klein four-group">Klein four-group</a> V as a proper <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a>, namely the even transpositions <span class="texhtml">{(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)},</span> with quotient S<sub>3</sub>. In <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, this map corresponds to the resolving cubic to a <a href="/wiki/Quartic_polynomial" class="mw-redirect" title="Quartic polynomial">quartic polynomial</a>, which allows the quartic to be solved by radicals, as established by <a href="/wiki/Lodovico_Ferrari" title="Lodovico Ferrari">Lodovico Ferrari</a>. The Klein group can be understood in terms of the <a href="/wiki/Lagrange_resolvent" class="mw-redirect" title="Lagrange resolvent">Lagrange resolvents</a> of the quartic. The map from S<sub>4</sub> to S<sub>3</sub> also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree <i>n</i> of dimension below <span class="texhtml"><i>n</i> − 1</span>, which only occurs for <span class="texhtml"><i>n</i> = 4</span>.</dd></dl> <dl><dt>S<sub>5</sub></dt> <dd>S<sub>5</sub> is the first non-solvable symmetric group. Along with the <a href="/wiki/Special_linear_group" title="Special linear group">special linear group</a> <span class="texhtml">SL(2, 5)</span> and the <a href="/wiki/Icosahedral_group" class="mw-redirect" title="Icosahedral group">icosahedral group</a> <span class="texhtml">A<sub>5</sub> × S<sub>2</sub></span>, S<sub>5</sub> is one of the three non-solvable groups of order 120, up to isomorphism. S<sub>5</sub> is the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of the general <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a>, and the fact that S<sub>5</sub> is not a <a href="/wiki/Solvable_group" title="Solvable group">solvable group</a> translates into the non-existence of a general formula to solve <a href="/wiki/Quintic_polynomial" class="mw-redirect" title="Quintic polynomial">quintic polynomials</a> by radicals. There is an exotic inclusion map <span class="texhtml">S<sub>5</sub> → S<sub>6</sub></span> as a <a href="#Transitive_subgroup_anchor">transitive subgroup</a>; the obvious inclusion map <span class="texhtml">S<sub><i>n</i></sub> → S<sub><i>n</i>+1</sub></span> fixes a point and thus is not transitive. This yields the outer automorphism of S<sub>6</sub>, discussed below, and corresponds to the resolvent sextic of a quintic.</dd></dl> <dl><dt>S<sub>6</sub></dt> <dd>Unlike all other symmetric groups, S<sub>6</sub>, has an <a href="/wiki/Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer automorphism</a>. Using the language of <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, this can also be understood in terms of <a href="/wiki/Lagrange_resolvents" class="mw-redirect" title="Lagrange resolvents">Lagrange resolvents</a>. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map <span class="texhtml">S<sub>5</sub> → S<sub>6</sub></span> as a transitive subgroup (the obvious inclusion map <span class="texhtml">S<sub><i>n</i></sub> → S<sub><i>n</i>+1</sub></span> fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of S<sub>6</sub>—see <i><a href="/wiki/Automorphisms_of_the_symmetric_and_alternating_groups" title="Automorphisms of the symmetric and alternating groups">Automorphisms of the symmetric and alternating groups</a></i> for details.</dd></dl> <dl><dd>Note that while A<sub>6</sub> and A<sub>7</sub> have an exceptional <a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a> (a <a href="/wiki/Covering_groups_of_the_alternating_and_symmetric_groups" title="Covering groups of the alternating and symmetric groups">triple cover</a>) and that these extend to triple covers of S<sub>6</sub> and S<sub>7</sub>, these do not correspond to exceptional Schur multipliers of the symmetric group.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Maps_between_symmetric_groups">Maps between symmetric groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=11" title="Edit section: Maps between symmetric groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Other than the trivial map <span class="texhtml">S<sub><i>n</i></sub> → C<sub>1</sub> ≅ S<sub>0</sub> ≅ S<sub>1</sub></span> and the sign map <span class="texhtml">S<sub><i>n</i></sub> → S<sub>2</sub></span>, the most notable homomorphisms between symmetric groups, in order of <a href="/wiki/Relative_dimension" title="Relative dimension">relative dimension</a>, are: </p> <ul><li><span class="texhtml">S<sub>4</sub> → S<sub>3</sub></span> corresponding to the exceptional normal subgroup <span class="texhtml">V < A<sub>4</sub> < S<sub>4</sub></span>;</li> <li><span class="texhtml">S<sub>6</sub> → S<sub>6</sub></span> (or rather, a class of such maps up to inner automorphism) corresponding to the outer automorphism of S<sub>6</sub>.</li> <li><span class="texhtml">S<sub>5</sub> → S<sub>6</sub></span> as a transitive subgroup, yielding the outer automorphism of S<sub>6</sub> as discussed above.</li></ul> <p>There are also a host of other homomorphisms <span class="texhtml">S<sub><i>m</i></sub> → S<sub><i>n</i></sub></span> where <span class="texhtml"><i>m</i> < <i>n</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Relation_with_alternating_group">Relation with alternating group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=12" title="Edit section: Relation with alternating group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For <span class="nowrap"><i>n</i> ≥ 5</span>, the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> A<sub><i>n</i></sub> is <a href="/wiki/Simple_group" title="Simple group">simple</a>, and the induced quotient is the sign map: <span class="nowrap">A<sub><i>n</i></sub> → S<sub><i>n</i></sub> → S<sub>2</sub></span> which is split by taking a transposition of two elements. Thus S<sub><i>n</i></sub> is the semidirect product <span class="nowrap">A<sub><i>n</i></sub> ⋊ S<sub>2</sub></span>, and has no other proper normal subgroups, as they would intersect A<sub><i>n</i></sub> in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in A<sub><i>n</i></sub> (and thus themselves be A<sub><i>n</i></sub> or S<sub><i>n</i></sub>). </p><p>S<sub><i>n</i></sub> acts on its subgroup A<sub><i>n</i></sub> by conjugation, and for <span class="nowrap"><i>n</i> ≠ 6</span>, S<sub><i>n</i></sub> is the full automorphism group of A<sub><i>n</i></sub>: Aut(A<sub><i>n</i></sub>) ≅ S<sub><i>n</i></sub>. Conjugation by even elements are <a href="/wiki/Inner_automorphism" title="Inner automorphism">inner automorphisms</a> of A<sub><i>n</i></sub> while the <a href="/wiki/Outer_automorphism" class="mw-redirect" title="Outer automorphism">outer automorphism</a> of A<sub><i>n</i></sub> of order 2 corresponds to conjugation by an odd element. For <span class="nowrap"><i>n</i> = 6</span>, there is an <a href="/wiki/Automorphisms_of_the_symmetric_and_alternating_groups#exceptional_outer_automorphism" title="Automorphisms of the symmetric and alternating groups">exceptional outer automorphism</a> of A<sub><i>n</i></sub> so S<sub><i>n</i></sub> is not the full automorphism group of A<sub><i>n</i></sub>. </p><p>Conversely, for <span class="nowrap"><i>n</i> ≠ 6</span>, S<sub><i>n</i></sub> has no outer automorphisms, and for <span class="nowrap"><i>n</i> ≠ 2</span> it has no center, so for <span class="nowrap"><i>n</i> ≠ 2, 6</span> it is a <a href="/wiki/Complete_group" title="Complete group">complete group</a>, as discussed in <a href="#Automorphism_group">automorphism group</a>, below. </p><p>For <span class="nowrap"><i>n</i> ≥ 5</span>, S<sub><i>n</i></sub> is an <a href="/wiki/Almost_simple_group" title="Almost simple group">almost simple group</a>, as it lies between the simple group A<sub><i>n</i></sub> and its group of automorphisms. </p><p>S<sub><i>n</i></sub> can be embedded into A<sub><i>n</i>+2</sub> by appending the transposition <span class="nowrap">(<i>n</i> + 1, <i>n</i> + 2)</span> to all odd permutations, while embedding into A<sub><i>n</i>+1</sub> is impossible for <span class="nowrap"><i>n</i> > 1</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Generators_and_relations">Generators and relations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=13" title="Edit section: Generators and relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The symmetric group on <span class="texhtml mvar" style="font-style:italic;">n</span> letters is generated by the <a href="/wiki/Adjacent_transposition" class="mw-redirect" title="Adjacent transposition">adjacent transpositions</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 \sigma _{i}=(i,i+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}=(i,i+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb828379fe56270619b12cc49994626f0b90f214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.677ex; height:2.843ex;" alt="{\displaystyle \sigma _{i}=(i,i+1)}"></span> that swap <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml"><i>i</i> + 1</span>.<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> The collection <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 \sigma _{1},\ldots ,\sigma _{n-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dd00945aa1aa2ee1c24d8312fa3c5fded9d53c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.206ex; height:2.009ex;" alt="{\displaystyle \sigma _{1},\ldots ,\sigma _{n-1}}"></span> generates <span class="texhtml">S<sub><i>n</i></sub></span> subject to the following relations:<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> <ul><li><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 \sigma _{i}^{2}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}^{2}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234fc4c0ed3f51720bc2acc2d0f3db0eb7de5385" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.292ex; height:3.176ex;" alt="{\displaystyle \sigma _{i}^{2}=1,}"></span></li> <li><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 \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708c02826f49f8a68897517be43947d257972333" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.827ex; height:2.343ex;" alt="{\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}}"></span> for <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 |i-j|>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>i</mi> <mo>−</mo> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |i-j|>1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4264363c0336c4efd00655cef76da743f5ac9914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.156ex; height:2.843ex;" alt="{\displaystyle |i-j|>1}"></span>, and</li> <li><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 (\sigma _{i}\sigma _{i+1})^{3}=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sigma _{i}\sigma _{i+1})^{3}=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdbb29ae8b7118b5323b659df44ae7135ea3dff2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.126ex; height:3.176ex;" alt="{\displaystyle (\sigma _{i}\sigma _{i+1})^{3}=1,}"></span></li></ul> <p>where 1 represents the identity permutation. This representation endows the symmetric group with the structure of a <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter group</a> (and so also a <a href="/wiki/Reflection_group" title="Reflection group">reflection group</a>). </p><p>Other possible generating sets include the set of transpositions that swap <span class="texhtml">1</span> and <span class="texhtml mvar" style="font-style:italic;">i</span> for <span class="texhtml">2 ≤ <i>i</i> ≤ <i>n</i></span>,<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> or more generally any set of transpositions that forms a connected graph,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> and a set containing any <span class="texhtml mvar" style="font-style:italic;">n</span>-cycle and a <span class="texhtml">2</span>-cycle of adjacent elements in the <span class="texhtml mvar" style="font-style:italic;">n</span>-cycle.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Subgroup_structure">Subgroup structure</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=14" title="Edit section: Subgroup structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of a symmetric group is called a <a href="/wiki/Permutation_group" title="Permutation group">permutation group</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Normal_subgroups">Normal subgroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=15" title="Edit section: Normal subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroups</a> of the finite symmetric groups are well understood. If <span class="texhtml"><i>n</i> ≤ 2</span>, S<sub><i>n</i></sub> has at most 2 elements, and so has no nontrivial proper subgroups. The <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> of degree <i>n</i> is always a normal subgroup, a proper one for <span class="texhtml"><i>n</i> ≥ 2</span> and nontrivial for <span class="texhtml"><i>n</i> ≥ 3</span>; for <span class="texhtml"><i>n</i> ≥ 3</span> it is in fact the only nontrivial proper normal subgroup of <span class="texhtml">S<sub><i>n</i></sub></span>, except when <span class="texhtml"><i>n</i> = 4</span> where there is one additional such normal subgroup, which is isomorphic to the <a href="/wiki/Klein_four_group" class="mw-redirect" title="Klein four group">Klein four group</a>. </p><p>The symmetric group on an infinite set does not have a subgroup of index 2, as <a href="/wiki/Giuseppe_Vitali" title="Giuseppe Vitali">Vitali</a> (1915<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>) proved that each permutation can be written as a product of three squares. (Any squared element must belong to the hypothesized subgroup of index 2, hence so must the product of any number of squares.) However it contains the normal subgroup <i>S</i> of permutations that fix all but finitely many elements, which is generated by transpositions. Those elements of <i>S</i> that are products of an even number of transpositions form a subgroup of index 2 in <i>S</i>, called the alternating subgroup <i>A</i>. Since <i>A</i> is even a <a href="/wiki/Characteristic_subgroup" title="Characteristic subgroup">characteristic subgroup</a> of <i>S</i>, it is also a normal subgroup of the full symmetric group of the infinite set. The groups <i>A</i> and <i>S</i> are the only nontrivial proper normal subgroups of the symmetric group on a countably infinite set. This was first proved by <a href="/w/index.php?title=Luigi_Onofri&action=edit&redlink=1" class="new" title="Luigi Onofri (page does not exist)">Onofri</a> (1929<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>) and independently <a href="/wiki/J%C3%B3zef_Schreier" title="Józef Schreier">Schreier</a>–<a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Ulam</a> (1934<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>). For more details see (<a href="#CITEREFScott1987">Scott 1987</a>, Ch. 11.3). That result, often called the Schreier-Ulam theorem, is superseded by a stronger one which says that the nontrivial normal subgroups of the symmetric group on a set <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> are 1) the even permutations with finite support and 2) for every cardinality <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 \aleph _{0}\leq \kappa \leq |X|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">ℵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≤</mo> <mi>κ</mi> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \aleph _{0}\leq \kappa \leq |X|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3712e03fb09e6628490c98e8e0c6dc4abe82c6a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.284ex; height:2.843ex;" alt="{\displaystyle \aleph _{0}\leq \kappa \leq |X|}"></span> the group of permutations with support less than <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 \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"></span> (<a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, Ch. 8.1). </p> <div class="mw-heading mw-heading3"><h3 id="Maximal_subgroups">Maximal subgroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=16" title="Edit section: Maximal subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Expand_section plainlinks metadata ambox mbox-small-left ambox-content" role="presentation"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w_cropped.svg" class="mw-file-description"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/20px-Wiki_letter_w_cropped.svg.png" decoding="async" width="20" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/30px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/40px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a></span></td><td class="mbox-text"><div class="mbox-text-span">This section <b>needs expansion</b>. You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Symmetric_group&action=edit&section=">adding to it</a>. <span class="date-container"><i>(<span class="date">September 2009</span>)</i></span></div></td></tr></tbody></table> <p>The <a href="/wiki/Maximal_subgroup" title="Maximal subgroup">maximal subgroups</a> of <span class="texhtml">S<sub><i>n</i></sub></span> fall into three classes: the intransitive, the imprimitive, and the primitive. The intransitive maximal subgroups are exactly those of the form <span class="texhtml">S<sub><i>k</i></sub> × S<sub><i>n</i>–<i>k</i></sub></span> for <span class="texhtml">1 ≤ <i>k</i> < <i>n</i>/2</span>. The imprimitive maximal subgroups are exactly those of the form <span class="texhtml">S<sub><i>k</i></sub> wr S<sub><i>n</i>/<i>k</i></sub></span>, where <span class="texhtml">2 ≤ <i>k</i> ≤ <i>n</i>/2</span> is a proper divisor of <i>n</i> and "wr" denotes the <a href="/wiki/Wreath_product" title="Wreath product">wreath product</a>. The primitive maximal subgroups are more difficult to identify, but with the assistance of the <a href="/wiki/O%27Nan%E2%80%93Scott_theorem" title="O'Nan–Scott theorem">O'Nan–Scott theorem</a> and the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>, (<a href="#CITEREFLiebeckPraegerSaxl1988">Liebeck, Praeger & Saxl 1988</a>) gave a fairly satisfactory description of the maximal subgroups of this type, according to (<a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 268). </p> <div class="mw-heading mw-heading3"><h3 id="Sylow_subgroups">Sylow subgroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=17" title="Edit section: Sylow subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Sylow_subgroup" class="mw-redirect" title="Sylow subgroup">Sylow subgroups</a> of the symmetric groups are important examples of <a href="/wiki/P-group" title="P-group"><i>p</i>-groups</a>. They are more easily described in special cases first: </p><p>The Sylow <i>p</i>-subgroups of the symmetric group of degree <i>p</i> are just the cyclic subgroups generated by <i>p</i>-cycles. There are <span class="texhtml">(<i>p</i> − 1)!/(<i>p</i> − 1) = (<i>p</i> − 2)!</span> such subgroups simply by counting <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">generators</a>. The <a href="/wiki/Normalizer" class="mw-redirect" title="Normalizer">normalizer</a> therefore has order <span class="texhtml"><i>p</i>⋅(<i>p</i> − 1)</span> and is known as a <a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a> <span class="texhtml"><i>F</i><sub><i>p</i>(<i>p</i>−1)</sub></span> (especially for <span class="texhtml"><i>p</i> = 5</span>), and is the <a href="/wiki/Affine_general_linear_group" class="mw-redirect" title="Affine general linear group">affine general linear group</a>, <span class="texhtml">AGL(1, <i>p</i>)</span>. </p><p>The Sylow <i>p</i>-subgroups of the symmetric group of degree <i>p</i><sup>2</sup> are the <a href="/wiki/Wreath_product" title="Wreath product">wreath product</a> of two cyclic groups of order <i>p</i>. For instance, when <span class="texhtml"><i>p</i> = 3</span>, a Sylow 3-subgroup of Sym(9) is generated by <span class="texhtml"><i>a</i> = (1 4 7)(2 5 8)(3 6 9)</span> and the elements <span class="texhtml"><i>x</i> = (1 2 3), <i>y</i> = (4 5 6), <i>z</i> = (7 8 9)</span>, and every element of the Sylow 3-subgroup has the form <span class="texhtml"><i>a</i><sup><i>i</i></sup><i>x</i><sup><i>j</i></sup><i>y</i><sup><i>k</i></sup><i>z</i><sup><i>l</i></sup></span> for <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq i,j,k,l\leq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo>≤</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq i,j,k,l\leq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf2f380c34e1c921304f8b20e2e75fc7e9f871e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.289ex; height:2.509ex;" alt="{\displaystyle 0\leq i,j,k,l\leq 2}"></span>⁠</span>. </p><p>The Sylow <i>p</i>-subgroups of the symmetric group of degree <i>p</i><sup><i>n</i></sup> are sometimes denoted W<sub><i>p</i></sub>(<i>n</i>), and using this notation one has that <span class="texhtml">W<sub><i>p</i></sub>(<i>n</i> + 1)</span> is the wreath product of W<sub><i>p</i></sub>(<i>n</i>) and W<sub><i>p</i></sub>(1). </p><p>In general, the Sylow <i>p</i>-subgroups of the symmetric group of degree <i>n</i> are a direct product of <i>a</i><sub><i>i</i></sub> copies of W<sub><i>p</i></sub>(<i>i</i>), where <span class="texhtml">0 ≤ <i>a<sub>i</sub></i> ≤ <i>p</i> − 1</span> and <span class="texhtml"><i>n</i> = <i>a</i><sub>0</sub> + <i>p</i>⋅<i>a</i><sub>1</sub> + ... + <i>p</i><sup><i>k</i></sup>⋅<i>a</i><sub><i>k</i></sub></span> (the base <i>p</i> expansion of <i>n</i>). </p><p>For instance, <span class="texhtml">W<sub>2</sub>(1) = C<sub>2</sub> and W<sub>2</sub>(2) = D<sub>8</sub></span>, the <a href="/wiki/Dihedral_group_of_order_8" title="Dihedral group of order 8">dihedral group of order 8</a>, and so a Sylow 2-subgroup of the <a class="mw-selflink selflink">symmetric group</a> of degree 7 is generated by <span class="texhtml">{ (1,3)(2,4), (1,2), (3,4), (5,6) } </span> and is isomorphic to <span class="texhtml">D<sub>8</sub> × C<sub>2</sub></span>. </p><p>These calculations are attributed to (<a href="#CITEREFKaloujnine1948">Kaloujnine 1948</a>) and described in more detail in (<a href="#CITEREFRotman1995">Rotman 1995</a>, p. 176). Note however that (<a href="#CITEREFKerber1971">Kerber 1971</a>, p. 26) attributes the result to an 1844 work of <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a>, and mentions that it is even covered in textbook form in (<a href="#CITEREFNetto1882">Netto 1882</a>, §39–40). </p> <div class="mw-heading mw-heading3"><h3 id="Transitive_subgroups"><span id="Transitive_subgroup_anchor"></span> Transitive subgroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=18" title="Edit section: Transitive subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>transitive subgroup</b> of S<sub><i>n</i></sub> is a subgroup whose action on {1, 2, ,..., <i>n</i>} is <a href="/wiki/Transitive_action" class="mw-redirect" title="Transitive action">transitive</a>. For example, the Galois group of a (<a href="/wiki/Finite_extension" class="mw-redirect" title="Finite extension">finite</a>) <a href="/wiki/Galois_extension" title="Galois extension">Galois extension</a> is a transitive subgroup of S<sub><i>n</i></sub>, for some <i>n</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Young_subgroups">Young subgroups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=19" title="Edit section: Young subgroups"><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/Young_subgroup" title="Young subgroup">Young subgroup</a></div> <p>A subgroup of <span class="texhtml">S<sub><i>n</i></sub></span> that is generated by transpositions is called a <i>Young subgroup</i>. They are all of the form <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_{a_{1}}\times \cdots \times S_{a_{\ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{a_{1}}\times \cdots \times S_{a_{\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91e750f6b8cfbc8a1f6de612650141ed6c46e51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.01ex; height:2.843ex;" alt="{\displaystyle S_{a_{1}}\times \cdots \times S_{a_{\ell }}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},\ldots ,a_{\ell })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},\ldots ,a_{\ell })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d15febf85107f2d96ea7d9ca1afccc79abec24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.419ex; height:2.843ex;" alt="{\displaystyle (a_{1},\ldots ,a_{\ell })}"></span> is an <a href="/wiki/Integer_partition" title="Integer partition">integer partition</a> of <span class="texhtml mvar" style="font-style:italic;">n</span>. These groups may also be characterized as the <a href="/wiki/Parabolic_subgroup_of_a_reflection_group" title="Parabolic subgroup of a reflection group">parabolic subgroups</a> of <span class="texhtml">S<sub><i>n</i></sub></span> when it is viewed as a <a href="/wiki/Reflection_group" title="Reflection group">reflection group</a>. </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=Symmetric_group&action=edit&section=20" title="Edit section: Cayley's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a> states that every group <i>G</i> is isomorphic to a subgroup of some symmetric group. In particular, one may take a subgroup of the symmetric group on the elements of <i>G</i>, since every group acts on itself faithfully by (left or right) multiplication. </p> <div class="mw-heading mw-heading2"><h2 id="Cyclic_subgroups">Cyclic subgroups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=21" title="Edit section: Cyclic subgroups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic groups</a> are those that are generated by a single permutation. When a permutation is represented in cycle notation, the order of the cyclic subgroup that it generates is the <a href="/wiki/Least_common_multiple" title="Least common multiple">least common multiple</a> of the lengths of its cycles. For example, in S<sub>5</sub>, one cyclic subgroup of order 5 is generated by (13254), whereas the largest cyclic subgroups of S<sub>5</sub> are generated by elements like (123)(45) that have one cycle of length 3 and another cycle of length 2. This rules out many groups as possible subgroups of symmetric groups of a given size.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2022)">citation needed</span></a></i>]</sup> For example, S<sub>5</sub> has no subgroup of order 15 (a divisor of the order of S<sub>5</sub>), because the only group of order 15 is the cyclic group. The largest possible order of a cyclic subgroup (equivalently, the largest possible order of an element in S<sub><i>n</i></sub>) is given by <a href="/wiki/Landau%27s_function" title="Landau's function">Landau's function</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Automorphism_group">Automorphism group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=22" title="Edit section: Automorphism group"><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">Further information: <a href="/wiki/Automorphisms_of_the_symmetric_and_alternating_groups" title="Automorphisms of the symmetric and alternating groups">Automorphisms of the symmetric and alternating groups</a></div> <table class="wikitable" style="float:right;" cellspacing="2"> <tbody><tr style="background:#a0e0a0;"> <td><i>n</i> </td> <td>Aut(S<sub><i>n</i></sub>) </td> <td>Out(S<sub><i>n</i></sub>) </td> <td>Z(S<sub><i>n</i></sub>) </td></tr> <tr> <td><i>n</i> ≠ 2, 6 </td> <td>S<sub><i>n</i></sub> </td> <td>C<sub>1</sub> </td> <td>C<sub>1</sub> </td></tr> <tr> <td><i>n</i> = 2 </td> <td>C<sub>1</sub> </td> <td>C<sub>1</sub> </td> <td>S<sub>2</sub> </td></tr> <tr> <td><i>n</i> = 6 </td> <td>S<sub>6</sub> ⋊ C<sub>2</sub> </td> <td>C<sub>2</sub> </td> <td>C<sub>1</sub> </td></tr></tbody></table> <p>For <span class="nowrap"><i>n</i> ≠ 2, 6</span>, S<sub><i>n</i></sub> is a <a href="/wiki/Complete_group" title="Complete group">complete group</a>: its <a href="/wiki/Center_(group_theory)" title="Center (group theory)">center</a> and <a href="/wiki/Outer_automorphism_group" title="Outer automorphism group">outer automorphism group</a> are both trivial. </p><p>For <span class="nowrap"><i>n</i> = 2</span>, the automorphism group is trivial, but S<sub>2</sub> is not trivial: it is isomorphic to C<sub>2</sub>, which is abelian, and hence the center is the whole group. </p><p>For <span class="nowrap"><i>n</i> = 6</span>, it has an outer automorphism of order 2: <span class="nowrap">Out(S<sub>6</sub>) = C<sub>2</sub></span>, and the automorphism group is a semidirect product <span class="nowrap">Aut(S<sub>6</sub>) = S<sub>6</sub> ⋊ C<sub>2</sub></span>. </p><p>In fact, for any set <i>X</i> of cardinality other than 6, every automorphism of the symmetric group on <i>X</i> is inner, a result first due to (<a href="#CITEREFSchreierUlam1936">Schreier & Ulam 1936</a>) according to (<a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 259). </p> <div class="mw-heading mw-heading2"><h2 id="Homology">Homology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=23" title="Edit section: Homology"><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">See also: <a href="/wiki/Alternating_group#Group_homology" title="Alternating group">Alternating group § Group homology</a></div> <p>The <a href="/wiki/Group_homology" class="mw-redirect" title="Group homology">group homology</a> of S<sub><i>n</i></sub> is quite regular and stabilizes: the first homology (concretely, the <a href="/wiki/Abelianization" class="mw-redirect" title="Abelianization">abelianization</a>) is: </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 H_{1}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<2\\\mathbf {Z} /2&n\geq 2.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <mo><</mo> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>n</mi> <mo>≥</mo> <mn>2.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<2\\\mathbf {Z} /2&n\geq 2.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f62b17d79ab0e4cd026c231361624033433efd27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.151ex; height:6.176ex;" alt="{\displaystyle H_{1}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<2\\\mathbf {Z} /2&n\geq 2.\end{cases}}}"></span></dd></dl> <p>The first homology group is the abelianization, and corresponds to the sign map S<sub><i>n</i></sub> → S<sub>2</sub> which is the abelianization for <i>n</i> ≥ 2; for <i>n</i> < 2 the symmetric group is trivial. This homology is easily computed as follows: S<sub><i>n</i></sub> is generated by involutions (2-cycles, which have order 2), so the only non-trivial maps <span class="nowrap">S<sub><i>n</i></sub> → C<sub><i>p</i></sub></span> are to S<sub>2</sub> and all involutions are conjugate, hence map to the same element in the abelianization (since conjugation is trivial in abelian groups). Thus the only possible maps <span class="nowrap">S<sub><i>n</i></sub> → S<sub>2</sub> ≅ {±1} </span> send an involution to 1 (the trivial map) or to −1 (the sign map). One must also show that the sign map is well-defined, but assuming that, this gives the first homology of S<sub><i>n</i></sub>. </p><p>The second homology (concretely, the <a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a>) is: </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 H_{2}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<4\\\mathbf {Z} /2&n\geq 4.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <mo><</mo> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mi>n</mi> <mo>≥</mo> <mn>4.</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<4\\\mathbf {Z} /2&n\geq 4.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d849927f3375640fba70bbd42b5ec8388c1581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.151ex; height:6.176ex;" alt="{\displaystyle H_{2}(\mathrm {S} _{n},\mathbf {Z} )={\begin{cases}0&n<4\\\mathbf {Z} /2&n\geq 4.\end{cases}}}"></span></dd></dl> <p>This was computed in (<a href="#CITEREFSchur1911">Schur 1911</a>), and corresponds to the <a href="/wiki/Covering_groups_of_the_alternating_and_symmetric_groups" title="Covering groups of the alternating and symmetric groups">double cover of the symmetric group</a>, 2 · S<sub><i>n</i></sub>. </p><p>Note that the <a href="/wiki/Exceptional_object" title="Exceptional object">exceptional</a> low-dimensional homology of the alternating 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 H_{1}(\mathrm {A} _{3})\cong H_{1}(\mathrm {A} _{4})\cong \mathrm {C} _{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≅</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}(\mathrm {A} _{3})\cong H_{1}(\mathrm {A} _{4})\cong \mathrm {C} _{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae82c40df4789ea154ce3c1643b4b47af8585649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.76ex; height:2.843ex;" alt="{\displaystyle H_{1}(\mathrm {A} _{3})\cong H_{1}(\mathrm {A} _{4})\cong \mathrm {C} _{3},}"></span> corresponding to non-trivial abelianization, 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 H_{2}(\mathrm {A} _{6})\cong H_{2}(\mathrm {A} _{7})\cong \mathrm {C} _{6},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≅</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{2}(\mathrm {A} _{6})\cong H_{2}(\mathrm {A} _{7})\cong \mathrm {C} _{6},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85932851f130d0199842fd8993d7aa466123ec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.76ex; height:2.843ex;" alt="{\displaystyle H_{2}(\mathrm {A} _{6})\cong H_{2}(\mathrm {A} _{7})\cong \mathrm {C} _{6},}"></span> due to the exceptional 3-fold cover) does not change the homology of the symmetric group; the alternating group phenomena do yield symmetric group phenomena – the map <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 \mathrm {A} _{4}\twoheadrightarrow \mathrm {C} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">↠</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {A} _{4}\twoheadrightarrow \mathrm {C} _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a142c31adde72cb72dd17b40bfdfa12d490211d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.144ex; height:2.509ex;" alt="{\displaystyle \mathrm {A} _{4}\twoheadrightarrow \mathrm {C} _{3}}"></span> extends 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 \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">↠</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3dea30d6e9b81e61dbe7819d846f2112b5c0496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.954ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3},}"></span> and the triple covers of A<sub>6</sub> and A<sub>7</sub> extend to triple covers of S<sub>6</sub> and S<sub>7</sub> – but these are not <i>homological</i> – the map <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 \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">↠</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c230a67385457833ca0d88ec07784bc4b32d1fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.308ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{4}\twoheadrightarrow \mathrm {S} _{3}}"></span> does not change the abelianization of S<sub>4</sub>, and the triple covers do not correspond to homology either. </p><p>The homology "stabilizes" in the sense of <a href="/wiki/Stable_homotopy" class="mw-redirect" title="Stable homotopy">stable homotopy</a> theory: there is an inclusion map <span class="nowrap">S<sub><i>n</i></sub> → S<sub><i>n</i>+1</sub></span>, and for fixed <i>k</i>, the induced map on homology <span class="nowrap"><i>H</i><sub><i>k</i></sub>(S<sub><i>n</i></sub>) → <i>H</i><sub><i>k</i></sub>(S<sub><i>n</i>+1</sub>)</span> is an isomorphism for sufficiently high <i>n</i>. This is analogous to the homology of families <a href="/wiki/Lie_groups" class="mw-redirect" title="Lie groups">Lie groups</a> stabilizing. </p><p>The homology of the infinite symmetric group is computed in (<a href="#CITEREFNakaoka1961">Nakaoka 1961</a>), with the cohomology algebra forming a <a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Representation_theory">Representation theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=24" title="Edit section: Representation theory"><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/Representation_theory_of_the_symmetric_group" title="Representation theory of the symmetric group">Representation theory of the symmetric group</a></div> <p>The <a href="/wiki/Representation_theory_of_the_symmetric_group" title="Representation theory of the symmetric group">representation theory of the symmetric group</a> is a particular case of the <a href="/wiki/Representation_theory_of_finite_groups" title="Representation theory of finite groups">representation theory of finite groups</a>, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from <a href="/wiki/Symmetric_function" title="Symmetric function">symmetric function</a> theory to problems of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> for a number of <a href="/wiki/Identical_particles" class="mw-redirect" title="Identical particles">identical particles</a>. </p><p>The symmetric group S<sub><i>n</i></sub> has order <i>n</i>!. Its <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy classes</a> are labeled by <a href="/wiki/Integer_partition" title="Integer partition">partitions</a> of <i>n</i>. Therefore, according to the representation theory of a finite group, the number of inequivalent <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible representations</a>, over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, is equal to the number of partitions of <i>n</i>. Unlike the general situation for finite groups, there is in fact a natural way to parametrize irreducible representation by the same set that parametrizes conjugacy classes, namely by partitions of <i>n</i> or equivalently <a href="/wiki/Young_diagram" class="mw-redirect" title="Young diagram">Young diagrams</a> of size <i>n</i>. </p><p>Each such irreducible representation can be realized over the integers (every permutation acting by a matrix with integer coefficients); it can be explicitly constructed by computing the <a href="/wiki/Young_symmetrizer" title="Young symmetrizer">Young symmetrizers</a> acting on a space generated by the <a href="/wiki/Young_tableau" title="Young tableau">Young tableaux</a> of shape given by the Young diagram. </p><p>Over other <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> the situation can become much more complicated. If the field <i>K</i> has <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> equal to zero or greater than <i>n</i> then by <a href="/wiki/Maschke%27s_theorem" title="Maschke's theorem">Maschke's theorem</a> the <a href="/wiki/Group_ring" title="Group ring">group algebra</a> <i>K</i>S<sub><i>n</i></sub> is semisimple. In these cases the irreducible representations defined over the integers give the complete set of irreducible representations (after reduction modulo the characteristic if necessary). </p><p>However, the irreducible representations of the symmetric group are not known in arbitrary characteristic. In this context it is more usual to use the language of <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> rather than representations. The representation obtained from an irreducible representation defined over the integers by reducing modulo the characteristic will not in general be irreducible. The modules so constructed are called <i><a href="/wiki/Specht_modules" class="mw-redirect" title="Specht modules">Specht modules</a></i>, and every irreducible does arise inside some such module. There are now fewer irreducibles, and although they can be classified they are very poorly understood. For example, even their <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimensions</a> are not known in general. </p><p>The determination of the irreducible modules for the symmetric group over an arbitrary field is widely regarded as one of the most important open problems in representation theory. </p> <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=Symmetric_group&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Braid_group" title="Braid group">Braid group</a></li> <li><a href="/wiki/History_of_group_theory" title="History of group theory">History of group theory</a></li> <li><a href="/wiki/Signed_symmetric_group" class="mw-redirect" title="Signed symmetric group">Signed symmetric group</a> and <a href="/wiki/Generalized_symmetric_group" title="Generalized symmetric group">Generalized symmetric group</a></li> <li><a href="/wiki/Symmetry_in_quantum_mechanics#Exchange_symmetry" title="Symmetry in quantum mechanics">Symmetry in quantum mechanics § Exchange symmetry</a></li> <li><a href="/wiki/Symmetric_inverse_semigroup" title="Symmetric inverse semigroup">Symmetric inverse semigroup</a></li> <li><a href="/wiki/Symmetric_power" title="Symmetric power">Symmetric power</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Symmetric_group&action=edit&section=26" 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 mw-references-columns"><ol class="references"> <li id="cite_note-Jacobson-def-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jacobson-def_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jacobson-def_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Jacobson-def_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Jacobson-def_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFJacobson2009">Jacobson 2009</a>, p. 31</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="#CITEREFJacobson2009">Jacobson 2009</a>, p. 32 Theorem 1.1</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"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Symmetric_Group_is_not_Abelian/Proof_1">"Symmetric Group is not Abelian/Proof 1"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Symmetric+Group+is+not+Abelian%2FProof+1&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FSymmetric_Group_is_not_Abelian%2FProof_1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></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="CITEREFVasishthaVasishtha2008" class="citation book cs1">Vasishtha, A.R.; Vasishtha, A.K. 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Krishna Prakashan Media. p. 49. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9788182830561" title="Special:BookSources/9788182830561"><bdi>9788182830561</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.+Groups+S3+Group+Definition&rft.btitle=Modern+Algebra&rft.pages=49&rft.pub=Krishna+Prakashan+Media&rft.date=2008&rft.isbn=9788182830561&rft.aulast=Vasishtha&rft.aufirst=A.R.&rft.au=Vasishtha%2C+A.K.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D45eCTUS6YnQC%26pg%3DPA49&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeubüser1967" class="citation thesis cs1">Neubüser, J. (1967). <i>Die Untergruppenverbände der Gruppen der Ordnungen ̤100 mit Ausnahme der Ordnungen 64 und 96</i> (PhD). Universität Kiel.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Die+Untergruppenverb%C3%A4nde+der+Gruppen+der+Ordnungen+%CC%A4100+mit+Ausnahme+der+Ordnungen+64+und+96&rft.inst=Universit%C3%A4t+Kiel&rft.date=1967&rft.aulast=Neub%C3%BCser&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></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="CITEREFSagan2001" class="citation cs2"><a href="/wiki/Bruce_Sagan" title="Bruce Sagan">Sagan, Bruce E.</a> (2001), <i>The Symmetric Group</i> (2 ed.), Springer, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Jm-HBaMdt8sC&pg=PA4">4</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-95067-9" title="Special:BookSources/978-0-387-95067-9"><bdi>978-0-387-95067-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Symmetric+Group&rft.pages=4&rft.edition=2&rft.pub=Springer&rft.date=2001&rft.isbn=978-0-387-95067-9&rft.aulast=Sagan&rft.aufirst=Bruce+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBjörnerBrenti2005" class="citation cs2"><a href="/wiki/Anders_Bj%C3%B6rner" title="Anders Björner">Björner, Anders</a>; Brenti, Francesco (2005), <i>Combinatorics of Coxeter groups</i>, Springer, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1TBPz5sd8m0C&pg=PA4">4. 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Rattan (2009), "Minimal factorizations of permutations into star transpositions", <i>Discrete Math.</i>, <b>309</b> (6): <span class="nowrap">1435–</span>1442, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.disc.2008.02.018">10.1016/j.disc.2008.02.018</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F96203">1721.1/96203</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Discrete+Math.&rft.atitle=Minimal+factorizations+of+permutations+into+star+transpositions&rft.volume=309&rft.issue=6&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1435-%3C%2Fspan%3E1442&rft.date=2009&rft_id=info%3Ahdl%2F1721.1%2F96203&rft_id=info%3Adoi%2F10.1016%2Fj.disc.2008.02.018&rft.au=J.+Irving&rft.au=A.+Rattan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTheo_DouvropoulosJoel_Brewster_LewisAlejandro_H._Morales2022" class="citation cs2">Theo Douvropoulos; Joel Brewster Lewis; Alejandro H. Morales (2022), "Hurwitz Numbers for Reflection Groups I: Generatingfunctionology", <i>Enumerative Combinatorics and Applications</i>, <b>2</b> (3), Proposition 2.1, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/2112.03427">2112.03427</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.54550%2FECA2022V2S3R20">10.54550/ECA2022V2S3R20</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Enumerative+Combinatorics+and+Applications&rft.atitle=Hurwitz+Numbers+for+Reflection+Groups+I%3A+Generatingfunctionology&rft.volume=2&rft.issue=3&rft.pages=Proposition+2.1&rft.date=2022&rft_id=info%3Aarxiv%2F2112.03427&rft_id=info%3Adoi%2F10.54550%2FECA2022V2S3R20&rft.au=Theo+Douvropoulos&rft.au=Joel+Brewster+Lewis&rft.au=Alejandro+H.+Morales&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin1991" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (1991), <i>Algebra</i>, Pearson, Exercise 6.6.16, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-004763-2" title="Special:BookSources/978-0-13-004763-2"><bdi>978-0-13-004763-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=Exercise+6.6.16&rft.pub=Pearson&rft.date=1991&rft.isbn=978-0-13-004763-2&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrayConderLeedham-GreenO'Brien2007" class="citation cs2">Bray, J.N.; Conder, M.D.E.; Leedham-Green, C.R.; O'Brien, E.A. (2007), <i>Short presentations for alternating and symmetric groups</i>, Transactions of the AMS</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Short+presentations+for+alternating+and+symmetric+groups&rft.pub=Transactions+of+the+AMS&rft.date=2007&rft.aulast=Bray&rft.aufirst=J.N.&rft.au=Conder%2C+M.D.E.&rft.au=Leedham-Green%2C+C.R.&rft.au=O%27Brien%2C+E.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVitali1915" class="citation journal cs1">Vitali, G. (1915). 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(1929). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02409971">"Teoria delle sostituzioni che operano su una infinità numerabile di elementi"</a>. <i>Annali di Matematica</i>. <b>7</b> (1): <span class="nowrap">103–</span>130. <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/10.1007%2FBF02409971">10.1007/BF02409971</a></span>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186219904">186219904</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annali+di+Matematica&rft.atitle=Teoria+delle+sostituzioni+che+operano+su+una+infinit%C3%A0+numerabile+di+elementi&rft.volume=7&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E103-%3C%2Fspan%3E130&rft.date=1929&rft_id=info%3Adoi%2F10.1007%2FBF02409971&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186219904%23id-name%3DS2CID&rft.aulast=Onofri&rft.aufirst=L.&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252FBF02409971&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchreierUlam1933" class="citation journal cs1">Schreier, J.; Ulam, S. (1933). <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/sm/sm4/sm4120.pdf">"Über die Permutationsgruppe der natürlichen Zahlenfolge"</a> <span class="cs1-format">(PDF)</span>. <i>Studia Math</i>. <b>4</b> (1): <span class="nowrap">134–</span>141. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Fsm-4-1-134-141">10.4064/sm-4-1-134-141</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studia+Math&rft.atitle=%C3%9Cber+die+Permutationsgruppe+der+nat%C3%BCrlichen+Zahlenfolge&rft.volume=4&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E134-%3C%2Fspan%3E141&rft.date=1933&rft_id=info%3Adoi%2F10.4064%2Fsm-4-1-134-141&rft.aulast=Schreier&rft.aufirst=J.&rft.au=Ulam%2C+S.&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Fsm%2Fsm4%2Fsm4120.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></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=Symmetric_group&action=edit&section=27" title="Edit section: References"><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="CITEREFCameron1999" class="citation cs2">Cameron, Peter J. (1999), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/permutationgroup0000came"><i>Permutation Groups</i></a></span>, London Mathematical Society Student Texts, vol. 45, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-65378-7" title="Special:BookSources/978-0-521-65378-7"><bdi>978-0-521-65378-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Permutation+Groups&rft.series=London+Mathematical+Society+Student+Texts&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-65378-7&rft.aulast=Cameron&rft.aufirst=Peter+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpermutationgroup0000came&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDixonMortimer1996" class="citation cs2">Dixon, John D.; Mortimer, Brian (1996), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/permutationgroup0000dixo"><i>Permutation groups</i></a></span>, Graduate Texts in Mathematics, vol. 163, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94599-6" title="Special:BookSources/978-0-387-94599-6"><bdi>978-0-387-94599-6</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1409812">1409812</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Permutation+groups&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1996&rft.isbn=978-0-387-94599-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1409812%23id-name%3DMR&rft.aulast=Dixon&rft.aufirst=John+D.&rft.au=Mortimer%2C+Brian&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpermutationgroup0000dixo&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson2009" class="citation cs2"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009), <i>Basic algebra</i>, vol. 1 (2nd ed.), Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47189-1" title="Special:BookSources/978-0-486-47189-1"><bdi>978-0-486-47189-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.edition=2nd&rft.pub=Dover&rft.date=2009&rft.isbn=978-0-486-47189-1&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaloujnine1948" class="citation cs2">Kaloujnine, Léo (1948), <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=ASENS_1948_3_65__239_0">"La structure des p-groupes de Sylow des groupes symétriques finis"</a>, <i><a href="/wiki/Annales_Scientifiques_de_l%27%C3%89cole_Normale_Sup%C3%A9rieure" title="Annales Scientifiques de l'École Normale Supérieure">Annales Scientifiques de l'École Normale Supérieure</a></i>, Série 3, <b>65</b>: <span class="nowrap">239–</span>276, <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/10.24033%2Fasens.961">10.24033/asens.961</a></span>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0012-9593">0012-9593</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=0028834">0028834</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&rft.atitle=La+structure+des+p-groupes+de+Sylow+des+groupes+sym%C3%A9triques+finis&rft.volume=65&rft.pages=%3Cspan+class%3D%22nowrap%22%3E239-%3C%2Fspan%3E276&rft.date=1948&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0028834%23id-name%3DMR&rft.issn=0012-9593&rft_id=info%3Adoi%2F10.24033%2Fasens.961&rft.aulast=Kaloujnine&rft.aufirst=L%C3%A9o&rft_id=http%3A%2F%2Fwww.numdam.org%2Fitem%3Fid%3DASENS_1948_3_65__239_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKerber1971" class="citation cs2">Kerber, Adalbert (1971), <i>Representations of permutation groups. I</i>, Lecture Notes in Mathematics, Vol. 240, vol. 240, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0067943">10.1007/BFb0067943</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-05693-5" title="Special:BookSources/978-3-540-05693-5"><bdi>978-3-540-05693-5</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=0325752">0325752</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Representations+of+permutation+groups.+I&rft.series=Lecture+Notes+in+Mathematics%2C+Vol.+240&rft.pub=Springer-Verlag&rft.date=1971&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0325752%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2FBFb0067943&rft.isbn=978-3-540-05693-5&rft.aulast=Kerber&rft.aufirst=Adalbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiebeckPraegerSaxl1988" class="citation cs2">Liebeck, M.W.; <a href="/wiki/Cheryl_Praeger" title="Cheryl Praeger">Praeger, C.E.</a>; <a href="/wiki/Jan_Saxl" title="Jan Saxl">Saxl, J.</a> (1988), "On the O'Nan–Scott theorem for finite primitive permutation groups", <i><a href="/wiki/Australian_Mathematical_Society#Society_journals" title="Australian Mathematical Society">Journal of the Australian Mathematical Society</a></i>, <b>44</b> (3): <span class="nowrap">389–</span>396, <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/10.1017%2FS144678870003216X">10.1017/S144678870003216X</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Australian+Mathematical+Society&rft.atitle=On+the+O%27Nan%E2%80%93Scott+theorem+for+finite+primitive+permutation+groups&rft.volume=44&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E389-%3C%2Fspan%3E396&rft.date=1988&rft_id=info%3Adoi%2F10.1017%2FS144678870003216X&rft.aulast=Liebeck&rft.aufirst=M.W.&rft.au=Praeger%2C+C.E.&rft.au=Saxl%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNakaoka1961" class="citation cs2">Nakaoka, Minoru (March 1961), "Homology of the Infinite Symmetric Group", <i><a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, 2, <b>73</b> (2): <span class="nowrap">229–</span>257, <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%2F1970333">10.2307/1970333</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/1970333">1970333</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Homology+of+the+Infinite+Symmetric+Group&rft.volume=73&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E229-%3C%2Fspan%3E257&rft.date=1961-03&rft_id=info%3Adoi%2F10.2307%2F1970333&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1970333%23id-name%3DJSTOR&rft.aulast=Nakaoka&rft.aufirst=Minoru&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNetto1882" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/Eugen_Netto" title="Eugen Netto">Netto, Eugen</a> (1882), <i>Substitutionentheorie und ihre Anwendungen auf die Algebra</i> (in German), Leipzig. Teubner, <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:14.0090.01">14.0090.01</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Substitutionentheorie+und+ihre+Anwendungen+auf+die+Algebra&rft.pub=Leipzig.+Teubner&rft.date=1882&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A14.0090.01%23id-name%3DJFM&rft.aulast=Netto&rft.aufirst=Eugen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman1995" class="citation cs2"><a href="/wiki/Joseph_J._Rotman" title="Joseph J. Rotman">Rotman, Joseph J.</a> (1995), <a rel="nofollow" class="external text" href="https://link.springer.com/content/pdf/10.1007%2F978-1-4612-4176-8_7.pdf">"Extensions and Cohomology"</a> <span class="cs1-format">(PDF)</span>, <i>An Introduction to the Theory of Groups</i>, Graduate Texts in Mathematics, vol. 148, Springer, pp. <span class="nowrap">154–</span>216, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-4176-8_7">10.1007/978-1-4612-4176-8_7</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4612-8686-8" title="Special:BookSources/978-1-4612-8686-8"><bdi>978-1-4612-8686-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Extensions+and+Cohomology&rft.btitle=An+Introduction+to+the+Theory+of+Groups&rft.series=Graduate+Texts+in+Mathematics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E154-%3C%2Fspan%3E216&rft.pub=Springer&rft.date=1995&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-4176-8_7&rft.isbn=978-1-4612-8686-8&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rft_id=https%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2F10.1007%252F978-1-4612-4176-8_7.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScott1987" class="citation cs2">Scott, W.R. (1987), <i>Group Theory</i>, <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>, pp. <span class="nowrap">45–</span>46, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-65377-8" title="Special:BookSources/978-0-486-65377-8"><bdi>978-0-486-65377-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory&rft.pages=%3Cspan+class%3D%22nowrap%22%3E45-%3C%2Fspan%3E46&rft.pub=Dover+Publications&rft.date=1987&rft.isbn=978-0-486-65377-8&rft.aulast=Scott&rft.aufirst=W.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchur1911" class="citation cs2"><a href="/wiki/Issai_Schur" title="Issai Schur">Schur, Issai</a> (1911), "Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen", <i><a href="/wiki/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a></i>, <b>1911</b> (139): <span class="nowrap">155–</span>250, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1911.139.155">10.1515/crll.1911.139.155</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122809608">122809608</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=%C3%9Cber+die+Darstellung+der+symmetrischen+und+der+alternierenden+Gruppe+durch+gebrochene+lineare+Substitutionen&rft.volume=1911&rft.issue=139&rft.pages=%3Cspan+class%3D%22nowrap%22%3E155-%3C%2Fspan%3E250&rft.date=1911&rft_id=info%3Adoi%2F10.1515%2Fcrll.1911.139.155&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122809608%23id-name%3DS2CID&rft.aulast=Schur&rft.aufirst=Issai&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchreierUlam1936" class="citation cs2 cs1-prop-foreign-lang-source"><a href="/wiki/J%C3%B3zef_Schreier" title="Józef Schreier">Schreier, Józef</a>; <a href="/wiki/Stanislaw_Ulam" class="mw-redirect" title="Stanislaw Ulam">Ulam, Stanislaw</a> (1936), <a rel="nofollow" class="external text" href="http://matwbn.icm.edu.pl/ksiazki/fm/fm28/fm28128.pdf">"Über die Automorphismen der Permutationsgruppe der natürlichen Zahlenfolge"</a> <span class="cs1-format">(PDF)</span>, <i><a href="/wiki/Fundamenta_Mathematicae" title="Fundamenta Mathematicae">Fundamenta Mathematicae</a></i> (in German), <b>28</b>: <span class="nowrap">258–</span>260, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4064%2Ffm-28-1-258-260">10.4064/fm-28-1-258-260</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:0016.20301">0016.20301</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Fundamenta+Mathematicae&rft.atitle=%C3%9Cber+die+Automorphismen+der+Permutationsgruppe+der+nat%C3%BCrlichen+Zahlenfolge&rft.volume=28&rft.pages=%3Cspan+class%3D%22nowrap%22%3E258-%3C%2Fspan%3E260&rft.date=1936&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0016.20301%23id-name%3DZbl&rft_id=info%3Adoi%2F10.4064%2Ffm-28-1-258-260&rft.aulast=Schreier&rft.aufirst=J%C3%B3zef&rft.au=Ulam%2C+Stanislaw&rft_id=http%3A%2F%2Fmatwbn.icm.edu.pl%2Fksiazki%2Ffm%2Ffm28%2Ffm28128.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+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=Symmetric_group&action=edit&section=28" title="Edit section: External links"><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 class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Symmetric_group">"Symmetric group"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Symmetric+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DSymmetric_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Symmetric_group"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SymmetricGroup.html">"Symmetric group"</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=Symmetric+group&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSymmetricGroup.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span></li> <li><span class="citation mathworld" id="Reference-Mathworld-Symmetric_group_graph"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20130624141646/http://mathworld.wolfram.com/SymmetricGroupGraph.html">"Symmetric group graph"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>. Archived from <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/SymmetricGroupGraph.html">the original</a> on 24 June 2013.</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=Symmetric+group+graph&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FSymmetricGroupGraph.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASymmetric+group" class="Z3988"></span></span></li> <li><a rel="nofollow" class="external text" href="http://www.ted.com/talks/marcus_du_sautoy_symmetry_reality_s_riddle.html">Marcus du Sautoy: Symmetry, reality's riddle</a> (video of a talk)</li> <li><a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a> <a rel="nofollow" class="external text" href="http://oeis.org/search?q=Symmetric+Group">Entries dealing with the Symmetric Group</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> Learning materials related to <a href="https://en.wikiversity.org/wiki/v:Symmetric_group_S4" class="extiw" title="v:v:Symmetric group S4">the S<sub>4</sub> symmetric group</a> at Wikiversity</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Symmetric_group&oldid=1275525027">https://en.wikipedia.org/w/index.php?title=Symmetric_group&oldid=1275525027</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Permutation_groups" title="Category:Permutation groups">Permutation groups</a></li><li><a href="/wiki/Category:Symmetry" title="Category:Symmetry">Symmetry</a></li><li><a href="/wiki/Category:Finite_reflection_groups" title="Category:Finite reflection groups">Finite reflection groups</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_September_2009" title="Category:Articles with unsourced statements from September 2009">Articles with unsourced statements from September 2009</a></li><li><a href="/wiki/Category:Articles_to_be_expanded_from_September_2009" title="Category:Articles to be expanded from September 2009">Articles to be expanded from September 2009</a></li><li><a href="/wiki/Category:All_articles_to_be_expanded" title="Category:All articles to be expanded">All articles to be expanded</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_October_2022" title="Category:Articles with unsourced statements from October 2022">Articles with unsourced statements from October 2022</a></li><li><a href="/wiki/Category:CS1_German-language_sources_(de)" title="Category:CS1 German-language sources (de)">CS1 German-language sources (de)</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 February 2025, at 15:23<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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