Permutation group - Wikipedia

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id="toc-Neutral_element_and_inverses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_actions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Group_actions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Group actions</span> </div> </a> <ul id="toc-Group_actions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transitive_actions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Transitive_actions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Transitive actions</span> </div> </a> <button aria-controls="toc-Transitive_actions-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 Transitive actions subsection</span> </button> <ul id="toc-Transitive_actions-sublist" class="vector-toc-list"> <li id="toc-Primitive_actions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primitive_actions"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Primitive actions</span> </div> </a> <ul id="toc-Primitive_actions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cayley's_theorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cayley's_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Cayley's theorem</span> </div> </a> <ul id="toc-Cayley's_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isomorphisms_of_permutation_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Isomorphisms_of_permutation_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Isomorphisms of permutation groups</span> </div> </a> <ul id="toc-Isomorphisms_of_permutation_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Oligomorphic_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Oligomorphic_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Oligomorphic groups</span> </div> </a> <ul id="toc-Oligomorphic_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span 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</div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">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 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class="mw-page-title-main">Permutation group</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 18 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-18" 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">18 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D8%AA%D8%A8%D8%AF%D9%8A%D9%84%D8%A7%D8%AA" title="زمرة تبديلات – Arabic" lang="ar" hreflang="ar" data-title="زمرة تبديلات" 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_de_permutacions" title="Grup de permutacions – Catalan" lang="ca" hreflang="ca" data-title="Grup de permutacions" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Permutationsgruppe" title="Permutationsgruppe – Danish" lang="da" hreflang="da" data-title="Permutationsgruppe" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Permutationsgruppe" title="Permutationsgruppe – German" lang="de" hreflang="de" data-title="Permutationsgruppe" 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_de_permutaciones" title="Grupo de permutaciones – Spanish" lang="es" hreflang="es" data-title="Grupo de permutaciones" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_de_permutations" title="Groupe de permutations – French" lang="fr" hreflang="fr" data-title="Groupe de permutations" 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_de_permutaci%C3%B3ns" title="Grupo de permutacións – Galician" lang="gl" hreflang="gl" data-title="Grupo de permutacións" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70893996 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EC%88%9C%EC%97%B4%EA%B5%B0" title="순열군 – Korean" lang="ko" hreflang="ko" data-title="순열군" 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_permutasi" title="Grup permutasi – Indonesian" lang="id" hreflang="id" data-title="Grup permutasi" 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-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Permutatiegroep" title="Permutatiegroep – Dutch" lang="nl" hreflang="nl" data-title="Permutatiegroep" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_permutacji" title="Grupa permutacji – Polish" lang="pl" hreflang="pl" data-title="Grupa permutacji" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_de_permuta%C3%A7%C3%A3o" title="Grupo de permutação – Portuguese" lang="pt" hreflang="pt" data-title="Grupo de permutação" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Permutaatioryhm%C3%A4" title="Permutaatioryhmä – Finnish" lang="fi" hreflang="fi" data-title="Permutaatioryhmä" 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/Permutationsgrupp" title="Permutationsgrupp – Swedish" lang="sv" hreflang="sv" data-title="Permutationsgrupp" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B0%E0%AE%BF%E0%AE%9A%E0%AF%88%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="வரிசைமாற்றுக் குலம் – Tamil" lang="ta" hreflang="ta" data-title="வரிசைமாற்றுக் குலம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D8%A8%D8%AF%D9%84_%DA%A9%D8%A7%D9%85%D9%84_%DA%AF%D8%B1%D9%88%DB%81" title="تبدل کامل گروہ – Urdu" lang="ur" hreflang="ur" data-title="تبدل کامل گروہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_ho%C3%A1n_v%E1%BB%8B" title="Nhóm hoán vị – Vietnamese" lang="vi" hreflang="vi" data-title="Nhóm hoán vị" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BD%AE%E6%8D%A2%E7%BE%A4" title="置换群 – Chinese" lang="zh" hreflang="zh" data-title="置换群" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div 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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 ); 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 mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li><li>SL(2, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Group_theory_sidebar" title="Template:Group theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Group_theory_sidebar" title="Template talk:Group theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Group_theory_sidebar" title="Special:EditPage/Template:Group theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>permutation group</b> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> whose elements are <a href="/wiki/Permutation" title="Permutation">permutations</a> of a given <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>M</i> and whose <a href="/wiki/Group_operation" class="mw-redirect" title="Group operation">group operation</a> is the composition of permutations in <i>G</i> (which are thought of as <a href="/wiki/Bijective_function" class="mw-redirect" title="Bijective function">bijective functions</a> from the set <i>M</i> to itself). The group of <i>all</i> permutations of a set <i>M</i> is the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> of <i>M</i>, often written as Sym(<i>M</i>).<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The term <i>permutation group</i> thus means a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the symmetric group. If <span class="nowrap"><i>M</i> = {1, 2, ..., <i>n</i>}</span> then Sym(<i>M</i>) is usually denoted by S<sub><i>n</i></sub>, and may be called the <i>symmetric group on n letters</i>. </p><p>By <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a>, every group is <a href="/wiki/Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to some permutation group. </p><p>The way in which the elements of a permutation group permute the elements of the set is called its <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a>. Group actions have applications in the study of <a href="/wiki/Symmetry" title="Symmetry">symmetries</a>, <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and many other branches of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Physics" title="Physics">physics</a> and chemistry. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Rubik%27s_cube.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/220px-Rubik%27s_cube.svg.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/330px-Rubik%27s_cube.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Rubik%27s_cube.svg/440px-Rubik%27s_cube.svg.png 2x" data-file-width="480" data-file-height="500" /></a><figcaption>The popular puzzle <a href="/wiki/Rubik%27s_cube" class="mw-redirect" title="Rubik's cube">Rubik's cube</a> invented in 1974 by <a href="/wiki/Ern%C5%91_Rubik" title="Ernő Rubik">Ernő Rubik</a> has been used as an illustration of permutation groups. Each rotation of a layer of the cube results in a <a href="/wiki/Permutation" title="Permutation">permutation</a> of the surface colors and is a member of the group. The permutation group of the cube is called the <a href="/wiki/Rubik%27s_cube_group" class="mw-redirect" title="Rubik's cube group">Rubik's cube group</a>.</figcaption></figure> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_properties_and_terminology">Basic properties and terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=1" title="Edit section: Basic properties and terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <i>permutation group</i> is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of a <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a>; that is, its elements are <a href="/wiki/Permutation" title="Permutation">permutations</a> of a given set. It is thus a subset of a symmetric group that is <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under <a href="/wiki/Function_composition" title="Function composition">composition</a> of permutations, contains the <a href="/wiki/Identity_permutation" class="mw-redirect" title="Identity permutation">identity permutation</a>, and contains the <a href="/wiki/Inverse_element" title="Inverse element">inverse permutation</a> of each of its elements.<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> A general property of finite groups implies that a finite nonempty subset of a symmetric group is a permutation group if and only if it is closed under permutation composition.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>The <b>degree</b> of a group of permutations of a <a href="/wiki/Finite_set" title="Finite set">finite set</a> is the <a href="/wiki/Cardinality" title="Cardinality">number of elements</a> in the set. The <b>order</b> of a group (of any type) is the number of elements (cardinality) in the group. By <a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a>, the order of any finite permutation group of degree <i>n</i> must divide <i>n</i>! since <i>n</i>-<a href="/wiki/Factorial" title="Factorial">factorial</a> is the order of the symmetric group <i>S</i><sub><i>n</i></sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=2" title="Edit section: Notation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Permutation#Notations" title="Permutation">Permutation § Notations</a></div> <p>Since permutations are <a href="/wiki/Bijection" title="Bijection">bijections</a> of a set, they can be represented by <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</a>'s <i>two-line notation</i>.<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> This notation lists each of the elements of <i>M</i> in the first row, and for each element, its image under the permutation below it in the second row. 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 \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> is a permutation of 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 M=\{x_{1},x_{2},\ldots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\{x_{1},x_{2},\ldots ,x_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028b3773fd27fe0502362eddc9a66ad3fe978d13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.394ex; height:2.843ex;" alt="{\displaystyle M=\{x_{1},x_{2},\ldots ,x_{n}\}}"></span> then, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\\sigma (x_{1})&\sigma (x_{2})&\sigma (x_{3})&\cdots &\sigma (x_{n})\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>σ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\\sigma (x_{1})&\sigma (x_{2})&\sigma (x_{3})&\cdots &\sigma (x_{n})\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb5f37a754f1c16d6078a22921b9795dba748ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:43.517ex; height:6.176ex;" alt="{\displaystyle \sigma ={\begin{pmatrix}x_{1}&x_{2}&x_{3}&\cdots &x_{n}\\\sigma (x_{1})&\sigma (x_{2})&\sigma (x_{3})&\cdots &\sigma (x_{n})\end{pmatrix}}.}"></span></dd></dl> <p>For instance, a particular permutation of the set {1, 2, 3, 4, 5} can be written as </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}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</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>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 \sigma ={\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/4942358c57e951a7fa882150a34c13d3acac573c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.35ex; height:6.176ex;" alt="{\displaystyle \sigma ={\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}};}"></span></dd></dl> <p>this means that <i>σ</i> satisfies <i>σ</i>(1) = 2, <i>σ</i>(2) = 5, <i>σ</i>(3) = 4, <i>σ</i>(4) = 3, and <i>σ</i>(5) = 1. The elements of <i>M</i> need not appear in any special order in the first row, so the same permutation could also be written as </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}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\begin{pmatrix}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ab1c2382c8426bd1bf2a4d0919a823fcff397b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.35ex; height:6.176ex;" alt="{\displaystyle \sigma ={\begin{pmatrix}3&2&5&1&4\\4&5&1&2&3\end{pmatrix}}.}"></span></dd></dl> <p>Permutations are also often written in <a href="/wiki/Cycle_notation" class="mw-redirect" title="Cycle notation">cycle notation</a> (<i>cyclic form</i>)<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> so that given the set <i>M</i> = {1, 2, 3, 4}, a permutation <i>g</i> of <i>M</i> with <i>g</i>(1) = 2, <i>g</i>(2) = 4, <i>g</i>(4) = 1 and <i>g</i>(3) = 3 will be written as (1, 2, 4)(3), or more commonly, (1, 2, 4) since 3 is left unchanged; if the objects are denoted by single letters or digits, commas and spaces can also be dispensed with, and we have a notation such as (124). The permutation written above in 2-line notation would be written in cycle notation as <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 =(125)(34).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>125</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>34</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =(125)(34).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dcfee85fb342462378f121ad83408684068eaae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.506ex; height:2.843ex;" alt="{\displaystyle \sigma =(125)(34).}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Composition_of_permutations–the_group_product"><span id="Composition_of_permutations.E2.80.93the_group_product"></span>Composition of permutations–the group product</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=3" title="Edit section: Composition of permutations–the group product"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The product of two permutations is defined as their <a href="/wiki/Function_composition" title="Function composition">composition</a> as functions, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma \cdot \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>⋅</mo> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \cdot \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a39d70c8aefb6a3929258f52324ff84975f23e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.341ex; height:1.676ex;" alt="{\displaystyle \sigma \cdot \pi }"></span> is the function that maps any element <i>x</i> of the set 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 \sigma (\pi (x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (\pi (x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ce1f3a34d3f6f8596151980f294b261b21fa86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.61ex; height:2.843ex;" alt="{\displaystyle \sigma (\pi (x))}"></span>. Note that the rightmost permutation is applied to the argument first, because of the way function composition is written.<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><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> Some authors prefer the leftmost factor acting first, but to that end permutations must be written to the <i>right</i> of their argument, often as a <a href="/wiki/Superscript" class="mw-redirect" title="Superscript">superscript</a>, so the 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> acting on the element <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/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> results in the image <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^{\sigma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\sigma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c644a7fade5b767896b4200586a54204016249d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.502ex; height:2.343ex;" alt="{\displaystyle x^{\sigma }}"></span>. With this convention, the product is given by <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^{\sigma \cdot \pi }=(x^{\sigma })^{\pi }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>⋅</mo> <mi>π</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{\sigma \cdot \pi }=(x^{\sigma })^{\pi }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02bf6f064d28aec71aa667a02880c9f627e28004" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.485ex; height:2.843ex;" alt="{\displaystyle x^{\sigma \cdot \pi }=(x^{\sigma })^{\pi }}"></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> <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> <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> However, this gives a <i>different</i> rule for multiplying permutations. This convention is commonly used in the permutation group literature, but this article uses the convention where the rightmost permutation is applied first. </p><p>Since the composition of two bijections always gives another bijection, the product of two permutations is again a permutation. In two-line notation, the product of two permutations is obtained by rearranging the columns of the second (leftmost) permutation so that its first row is identical with the second row of the first (rightmost) permutation. The product can then be written as the first row of the first permutation over the second row of the modified second permutation. For example, given the permutations, </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 P={\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}\quad {\text{ and }}\quad Q={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</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>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mspace width="1em" /> <mi>Q</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>5</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}\quad {\text{ and }}\quad Q={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24ca0694573e4eea9cb94d1bd13cd3f96430f04a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:58.532ex; height:6.176ex;" alt="{\displaystyle P={\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}\quad {\text{ and }}\quad Q={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}},}"></span></dd></dl> <p>the product <i>QP</i> 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 QP={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}2&4&1&3&5\\4&2&5&3&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\4&2&5&3&1\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mi>P</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>5</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <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>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <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>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <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>5</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 QP={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}2&4&1&3&5\\4&2&5&3&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\4&2&5&3&1\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9318d868c152458fd2e03346cbaacca1800c697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:109.902ex; height:6.176ex;" alt="{\displaystyle QP={\begin{pmatrix}1&2&3&4&5\\5&4&3&2&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}2&4&1&3&5\\4&2&5&3&1\end{pmatrix}}{\begin{pmatrix}1&2&3&4&5\\2&4&1&3&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\4&2&5&3&1\end{pmatrix}}.}"></span></dd></dl> <p>The composition of permutations, when they are written in cycle notation, is obtained by juxtaposing the two permutations (with the second one written on the left) and then simplifying to a disjoint cycle form if desired. Thus, the above product would be 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 Q\cdot P=(15)(24)\cdot (1243)=(1435).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>⋅</mo> <mi>P</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>15</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>24</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1243</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1435</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q\cdot P=(15)(24)\cdot (1243)=(1435).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85554bb0cf6aea986db25aced8a8b36298ee219c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.972ex; height:2.843ex;" alt="{\displaystyle Q\cdot P=(15)(24)\cdot (1243)=(1435).}"></span></dd></dl> <p>Since function composition is <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, so is the product operation on permutations: <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 \cdot \pi )\cdot \rho =\sigma \cdot (\pi \cdot \rho )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>⋅</mo> <mi>π</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>ρ</mi> <mo>=</mo> <mi>σ</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>π</mi> <mo>⋅</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sigma \cdot \pi )\cdot \rho =\sigma \cdot (\pi \cdot \rho )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f506eec5d7840b778ed578ce06fb22418346e035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.161ex; height:2.843ex;" alt="{\displaystyle (\sigma \cdot \pi )\cdot \rho =\sigma \cdot (\pi \cdot \rho )}"></span>. Therefore, products of two or more permutations are usually written without adding parentheses to express grouping; they are also usually written without a dot or other sign to indicate multiplication (the dots of the previous example were added for emphasis, so would simply be written as <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 \pi \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mi>π</mi> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \pi \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2529f90eb31d8c60e80a8f9e3b1910d6324aeeaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.864ex; height:2.176ex;" alt="{\displaystyle \sigma \pi \rho }"></span>). </p> <div class="mw-heading mw-heading2"><h2 id="Neutral_element_and_inverses">Neutral element and inverses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=4" title="Edit section: Neutral element and inverses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The identity permutation, which maps every element of the set to itself, is the <a href="/wiki/Neutral_element" class="mw-redirect" title="Neutral element">neutral element</a> for this product. In two-line notation, the identity 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 {\begin{pmatrix}1&2&3&\cdots &n\\1&2&3&\cdots &n\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> <mn>3</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>n</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&2&3&\cdots &n\\1&2&3&\cdots &n\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44788279d909fb44f13ccb9a3f294093b5538d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.715ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&2&3&\cdots &n\\1&2&3&\cdots &n\end{pmatrix}}.}"></span></dd></dl> <p>In cycle notation, <i>e</i> = (1)(2)(3)...(<i>n</i>) which by convention is also denoted by just (1) or even ().<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><p>Since <a href="/wiki/Bijections" class="mw-redirect" title="Bijections">bijections</a> have <a href="/wiki/Inverse_function" title="Inverse function">inverses</a>, so do permutations, and the inverse <i>σ</i><sup>−1</sup> of <i>σ</i> is again a permutation. Explicitly, whenever <i>σ</i>(<i>x</i>)=<i>y</i> one also has <i>σ</i><sup>−1</sup>(<i>y</i>)=<i>x</i>. In two-line notation the inverse can be obtained by interchanging the two lines (and sorting the columns if one wishes the first line to be in a given order). For instance </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&3&4&5\\2&5&4&3&1\end{pmatrix}}^{-1}={\begin{pmatrix}2&5&4&3&1\\1&2&3&4&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\5&1&4&3&2\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <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> <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> <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> <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> </mtable> <mo>)</mo> </mrow> </mrow> <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>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}^{-1}={\begin{pmatrix}2&5&4&3&1\\1&2&3&4&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\5&1&4&3&2\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff21f9d8c52c3074fa1c784dd985f9b788719082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.002ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}1&2&3&4&5\\2&5&4&3&1\end{pmatrix}}^{-1}={\begin{pmatrix}2&5&4&3&1\\1&2&3&4&5\end{pmatrix}}={\begin{pmatrix}1&2&3&4&5\\5&1&4&3&2\end{pmatrix}}.}"></span></dd></dl> <p>To obtain the inverse of a single cycle, we reverse the order of its elements. Thus, </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 (125)^{-1}=(521)=(152).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>125</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>521</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>152</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (125)^{-1}=(521)=(152).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf235c210c006f9f70247ef832fba121bcda384" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.067ex; height:3.176ex;" alt="{\displaystyle (125)^{-1}=(521)=(152).}"></span></dd></dl> <p>To obtain the inverse of a product of cycles, we first reverse the order of the cycles, and then we take the inverse of each as above. Thus, </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 [(125)(34)]^{-1}=(34)^{-1}(125)^{-1}=(43)(521)=(34)(152).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mn>125</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>34</mn> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>34</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>125</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>43</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>521</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>34</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>152</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [(125)(34)]^{-1}=(34)^{-1}(125)^{-1}=(43)(521)=(34)(152).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60acfa818803c852ee3949bda58e7e31d72ffa2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.958ex; height:3.176ex;" alt="{\displaystyle [(125)(34)]^{-1}=(34)^{-1}(125)^{-1}=(43)(521)=(34)(152).}"></span></dd></dl> <p>Having an associative product, an identity element, and inverses for all its elements, makes the set of all permutations of <i>M</i> into a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>, Sym(<i>M</i>); a permutation group. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=5" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the following set <i>G</i><sub>1</sub> of permutations of the set <i>M</i> = {1, 2, 3, 4}: </p> <ul><li><i>e</i> = (1)(2)(3)(4) = (1) <ul><li>This is the identity, the trivial permutation which fixes each element.</li></ul></li> <li><i>a</i> = (1 2)(3)(4) = (1 2) <ul><li>This permutation interchanges 1 and 2, and fixes 3 and 4.</li></ul></li> <li><i>b</i> = (1)(2)(3 4) = (3 4) <ul><li>Like the previous one, but exchanging 3 and 4, and fixing the others.</li></ul></li> <li><i>ab</i> = (1 2)(3 4) <ul><li>This permutation, which is the composition of the previous two, exchanges simultaneously 1 with 2, and 3 with 4.</li></ul></li></ul> <p><i>G</i><sub>1</sub> forms a group, since <i>aa</i> = <i>bb</i> = <i>e</i>, <i>ba</i> = <i>ab</i>, and <i>abab</i> = <i>e</i>. This permutation group is, as an <a href="/wiki/Abstract_group" class="mw-redirect" title="Abstract group">abstract group</a>, the <a href="/wiki/Klein_group" class="mw-redirect" title="Klein group">Klein group</a> <i>V</i><sub>4</sub>. </p><p>As another example consider the <a href="/wiki/Examples_of_groups#The_symmetry_group_of_a_square:_dihedral_group_of_order_8" title="Examples of groups">group of symmetries of a square</a>. Let the vertices of a square be labeled 1, 2, 3 and 4 (counterclockwise around the square starting with 1 in the top left corner). The symmetries are determined by the images of the vertices, that can, in turn, be described by permutations. The rotation by 90° (counterclockwise) about the center of the square is described by the permutation (1234). The 180° and 270° rotations are given by (13)(24) and (1432), respectively. The reflection about the horizontal line through the center is given by (12)(34) and the corresponding vertical line reflection is (14)(23). The reflection about the 1,3−diagonal line is (24) and reflection about the 2,4−diagonal is (13). The only remaining symmetry is the identity (1)(2)(3)(4). This permutation group is known, as an abstract group, as the <a href="/wiki/Dihedral_group" title="Dihedral group">dihedral group</a> of order 8. </p> <div class="mw-heading mw-heading2"><h2 id="Group_actions">Group actions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=6" title="Edit section: Group actions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">Group action (mathematics)</a></div> <p>In the above example of the symmetry group of a square, the permutations "describe" the movement of the vertices of the square induced by the group of symmetries. It is common to say that these group elements are "acting" on the set of vertices of the square. This idea can be made precise by formally defining a <b>group action</b>.<sup id="cite_ref-Dixon96_12-0" class="reference"><a href="#cite_note-Dixon96-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>Let <i>G</i> be a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and <i>M</i> a nonempty <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>. An <b>action</b> of <i>G</i> on <i>M</i> is a function <i>f</i>: <i>G</i> × <i>M</i> → <i>M</i> such that </p> <ul><li><i>f</i>(1, <i>x</i>) = <i>x</i>, for all <i>x</i> in <i>M</i> (1 is the <a href="/wiki/Identity_element" title="Identity element">identity</a> (neutral) element of the group <i>G</i>), and</li> <li><i>f</i>(<i>g</i>, <i>f</i>(<i>h</i>, <i>x</i>)) = <i>f</i>(<i>gh</i>, <i>x</i>), for all <i>g</i>,<i>h</i> in <i>G</i> and all <i>x</i> in <i>M</i>.</li></ul> <p>This pair of conditions can also be expressed as saying that the action induces a group homomorphism from <i>G</i> into <i>Sym</i>(<i>M</i>).<sup id="cite_ref-Dixon96_12-1" class="reference"><a href="#cite_note-Dixon96-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Any such homomorphism is called a <i>(permutation) representation</i> of <i>G</i> on <i>M</i>. </p><p>For any permutation group, the action that sends (<i>g</i>, <i>x</i>) → <i>g</i>(<i>x</i>) is called the <b>natural action</b> of <i>G</i> on <i>M</i>. This is the action that is assumed unless otherwise indicated.<sup id="cite_ref-Dixon96_12-2" class="reference"><a href="#cite_note-Dixon96-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In the example of the symmetry group of the square, the group's action on the set of vertices is the natural action. However, this group also induces an action on the set of four triangles in the square, which are: <i>t</i><sub>1</sub> = 234, <i>t</i><sub>2</sub> = 134, <i>t</i><sub>3</sub> = 124 and <i>t</i><sub>4</sub> = 123. It also acts on the two diagonals: <i>d</i><sub>1</sub> = 13 and <i>d</i><sub>2</sub> = 24. </p> <table class="wikitable"> <tbody><tr> <th>Group element</th> <th>Action on triangles</th> <th>Action on diagonals </th></tr> <tr> <td>(1)</td> <td>(1)</td> <td>(1) </td></tr> <tr> <td>(1234)</td> <td>(<i>t</i><sub>1</sub> <i>t</i><sub>2</sub> <i>t</i><sub>3</sub> <i>t</i><sub>4</sub>)</td> <td>(<i>d</i><sub>1</sub> <i>d</i><sub>2</sub>) </td></tr> <tr> <td>(13)(24)</td> <td>(<i>t</i><sub>1</sub> <i>t</i><sub>3</sub>)(<i>t</i><sub>2</sub> <i>t</i><sub>4</sub>)</td> <td>(1) </td></tr> <tr> <td>(1432)</td> <td>(<i>t</i><sub>1</sub> <i>t</i><sub>4</sub> <i>t</i><sub>3</sub> <i>t</i><sub>2</sub>)</td> <td>(<i>d</i><sub>1</sub> <i>d</i><sub>2</sub>) </td></tr> <tr> <td>(12)(34)</td> <td>(<i>t</i><sub>1</sub> <i>t</i><sub>2</sub>)(<i>t</i><sub>3</sub> <i>t</i><sub>4</sub>)</td> <td>(<i>d</i><sub>1</sub> <i>d</i><sub>2</sub>) </td></tr> <tr> <td>(14)(23)</td> <td>(<i>t</i><sub>1</sub> <i>t</i><sub>4</sub>)(<i>t</i><sub>2</sub> <i>t</i><sub>3</sub>)</td> <td>(<i>d</i><sub>1</sub> <i>d</i><sub>2</sub>) </td></tr> <tr> <td>(13)</td> <td>(<i>t</i><sub>1</sub> <i>t</i><sub>3</sub>)</td> <td>(1) </td></tr> <tr> <td>(24)</td> <td>(<i>t</i><sub>2</sub> <i>t</i><sub>4</sub>)</td> <td>(1) </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Transitive_actions">Transitive actions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=7" title="Edit section: Transitive actions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The action of a group <i>G</i> on a set <i>M</i> is said to be <i>transitive</i> if, for every two elements <i>s</i>, <i>t</i> of <i>M</i>, there is some group element <i>g</i> such that <i>g</i>(<i>s</i>) = <i>t</i>. Equivalently, the set <i>M</i> forms a single <a href="/wiki/Orbit_(group_theory)" class="mw-redirect" title="Orbit (group theory)">orbit</a> under the action of <i>G</i>.<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> Of the examples <a href="#Examples">above</a>, the group {e, (1 2), (3 4), (1 2)(3 4)} of permutations of {1, 2, 3, 4} is not transitive (no group element takes 1 to 3) but the group of symmetries of a square is transitive on the vertices. </p> <div class="mw-heading mw-heading3"><h3 id="Primitive_actions">Primitive actions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=8" title="Edit section: Primitive actions"><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/Primitive_permutation_group" title="Primitive permutation group">Primitive permutation group</a></div> <p>A permutation group <i>G</i> acting transitively on a non-empty finite set <i>M</i> is <i>imprimitive</i> if there is some nontrivial <a href="/wiki/Set_partition" class="mw-redirect" title="Set partition">set partition</a> of <i>M</i> that is preserved by the action of <i>G</i>, where "nontrivial" means that the partition isn't the partition into <a href="/wiki/Singleton_set" class="mw-redirect" title="Singleton set">singleton sets</a> nor the partition with only one part. Otherwise, if <i>G</i> is transitive but does not preserve any nontrivial partition of <i>M</i>, the group <i>G</i> is <i>primitive</i>. </p><p>For example, the group of symmetries of a square is imprimitive on the vertices: if they are numbered 1, 2, 3, 4 in cyclic order, then the partition {{1, 3}, {2, 4}} into opposite pairs is preserved by every group element. On the other hand, the full symmetric group on a set <i>M</i> is always primitive. </p> <div class="mw-heading mw-heading2"><h2 id="Cayley's_theorem"><span id="Cayley.27s_theorem"></span>Cayley's theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=9" title="Edit section: Cayley's theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a></div> <p>Any group <i>G</i> can act on itself (the elements of the group being thought of as the set <i>M</i>) in many ways. In particular, there is a <a href="/wiki/Regular_group_action" class="mw-redirect" title="Regular group action">regular action</a> given by (left) multiplication in the group. That is, <i>f</i>(<i>g</i>, <i>x</i>) = <i>gx</i> for all <i>g</i> and <i>x</i> in <i>G</i>. For each fixed <i>g</i>, the function <i>f</i><sub><i>g</i></sub>(<i>x</i>) = <i>gx</i> is a bijection on <i>G</i> and therefore a permutation of the set of elements of <i>G</i>. Each element of <i>G</i> can be thought of as a permutation in this way and so <i>G</i> is isomorphic to a permutation group; this is the content of <a href="/wiki/Cayley%27s_theorem" title="Cayley's theorem">Cayley's theorem</a>. </p><p>For example, consider the group <i>G</i><sub>1</sub> acting on the set {1, 2, 3, 4} given above. Let the elements of this group be denoted by <i>e</i>, <i>a</i>, <i>b</i> and <i>c</i> = <i>ab</i> = <i>ba</i>. The action of <i>G</i><sub>1</sub> on itself described in Cayley's theorem gives the following permutation representation: </p> <dl><dd><i>f</i><sub><i>e</i></sub> ↦ (<i>e</i>)(<i>a</i>)(<i>b</i>)(<i>c</i>)</dd> <dd><i>f</i><sub><i>a</i></sub> ↦ (<i>ea</i>)(<i>bc</i>)</dd> <dd><i>f</i><sub><i>b</i></sub> ↦ (<i>eb</i>)(<i>ac</i>)</dd> <dd><i>f</i><sub><i>c</i></sub> ↦ (<i>ec</i>)(<i>ab</i>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Isomorphisms_of_permutation_groups">Isomorphisms of permutation groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=10" title="Edit section: Isomorphisms of permutation groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>G</i> and <i>H</i> are two permutation groups on sets <i>X</i> and <i>Y</i> with actions <i>f</i><sub>1</sub> and <i>f</i><sub>2</sub> respectively, then we say that <i>G</i> and <i>H</i> are <i>permutation isomorphic</i> (or <i><a href="/wiki/Isomorphism" title="Isomorphism">isomorphic</a> as permutation groups</i>) if there exists a <a href="/wiki/Bijection" title="Bijection">bijective map</a> <span class="nowrap"><i>λ</i> : <i>X</i> → <i>Y</i></span> and a <a href="/wiki/Group_isomorphism" title="Group isomorphism">group isomorphism</a> <span class="nowrap"><i>ψ</i> : <i>G</i> → <i>H</i></span> such that </p> <dl><dd><i>λ</i>(<i>f</i><sub>1</sub>(<i>g</i>, <i>x</i>)) = <i>f</i><sub>2</sub>(<i>ψ</i>(<i>g</i>), <i>λ</i>(<i>x</i>)) for all <i>g</i> in <i>G</i> and <i>x</i> in <i>X</i>.<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></dd></dl> <p>If <span class="nowrap"><i>X</i> = <i>Y</i></span> this is equivalent to <i>G</i> and <i>H</i> being conjugate as subgroups of Sym(<i>X</i>).<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The special case where <span class="nowrap"><i>G</i> = <i>H</i></span> and <i>ψ</i> is the <a href="/wiki/Identity_map" class="mw-redirect" title="Identity map">identity map</a> gives rise to the concept of <i>equivalent actions</i> of a group.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>In the example of the symmetries of a square given above, the natural action on the set {1,2,3,4} is equivalent to the action on the triangles. The bijection <i>λ</i> between the sets is given by <span class="nowrap"><i>i</i> ↦ <i>t</i><sub><i>i</i></sub></span>. The natural action of group <i>G</i><sub>1</sub> above and its action on itself (via left multiplication) are not equivalent as the natural action has fixed points and the second action does not. </p> <div class="mw-heading mw-heading2"><h2 id="Oligomorphic_groups">Oligomorphic groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=11" title="Edit section: Oligomorphic groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When a group <i>G</i> acts on a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> <i>S</i>, the action may be extended naturally to the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <i>S<sup>n</sup></i> of <i>S</i>, consisting of <i>n</i>-tuples of elements of <i>S</i>: the action of an element <i>g</i> on the <i>n</i>-tuple (<i>s</i><sub>1</sub>, ..., <i>s</i><sub><i>n</i></sub>) is given by </p> <dl><dd><i>g</i>(<i>s</i><sub>1</sub>, ..., <i>s</i><sub><i>n</i></sub>) = (<i>g</i>(<i>s</i><sub>1</sub>), ..., <i>g</i>(<i>s</i><sub><i>n</i></sub>)).</dd></dl> <p>The group <i>G</i> is said to be <i>oligomorphic</i> if the action on <i>S<sup>n</sup></i> has only finitely many orbits for every positive integer <i>n</i>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> (This is automatic if <i>S</i> is finite, so the term is typically of interest when <i>S</i> is infinite.) </p><p>The interest in oligomorphic groups is partly based on their application to <a href="/wiki/Model_theory" title="Model theory">model theory</a>, for example when considering <a href="/wiki/Automorphism" title="Automorphism">automorphisms</a> in <a href="/wiki/Countably_categorical_theory" class="mw-redirect" title="Countably categorical theory">countably categorical theories</a>.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=12" title="Edit section: History"><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/History_of_group_theory" title="History of group theory">History of group theory</a></div> <p>The study of <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> originally grew out of an understanding of permutation groups.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Permutation" title="Permutation">Permutations</a> had themselves been intensively studied by <a href="/wiki/Lagrange" class="mw-redirect" title="Lagrange">Lagrange</a> in 1770 in his work on the algebraic solutions of polynomial equations. This subject flourished and by the mid 19th century a well-developed theory of permutation groups existed, codified by <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a> in his book <i>Traité des Substitutions et des Équations Algébriques</i> of 1870. Jordan's book was, in turn, based on the papers that were left by <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> in 1832. </p><p>When <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> introduced the concept of an <a href="/wiki/Abstract_group" class="mw-redirect" title="Abstract group">abstract group</a>, it was not immediately clear whether or not this was a larger collection of objects than the known permutation groups (which had a definition different from the modern one). Cayley went on to prove that the two concepts were equivalent in Cayley's theorem.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p><p>Another classical text containing several chapters on permutation groups is <a href="/wiki/William_Burnside" title="William Burnside">Burnside</a>'s <i>Theory of Groups of Finite Order</i> of 1911.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The first half of the twentieth century was a fallow period in the study of group theory in general, but interest in permutation groups was revived in the 1950s by <a href="/wiki/H._Wielandt" class="mw-redirect" title="H. Wielandt">H. Wielandt</a> whose German lecture notes were reprinted as <i>Finite Permutation Groups</i> in 1964.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </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=Permutation_group&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/2-transitive_group" class="mw-redirect" title="2-transitive group">2-transitive group</a></li> <li><a href="/wiki/Rank_3_permutation_group" title="Rank 3 permutation group">Rank 3 permutation group</a></li> <li><a href="/wiki/Mathieu_group" title="Mathieu group">Mathieu group</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=Permutation_group&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The notations <b>S</b><sub><i>M</i></sub> and <b>S</b><sup><i>M</i></sup> are also used.</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="#CITEREFRotman2006">Rotman 2006</a>, p. 148, Definition of subgroup</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFRotman2006">Rotman 2006</a>, p. 149, Proposition 2.69</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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWussing2007" class="citation cs2">Wussing, Hans (2007), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Xp3JymnfAq4C&pg=PA94"><i>The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory</i></a>, Courier Dover Publications, p. 94, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486458687" title="Special:BookSources/9780486458687"><bdi>9780486458687</bdi></a>, <q>Cauchy used his permutation notation—in which the arrangements are written one below the other and both are enclosed in parentheses—for the first time in 1815.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Genesis+of+the+Abstract+Group+Concept%3A+A+Contribution+to+the+History+of+the+Origin+of+Abstract+Group+Theory&rft.pages=94&rft.pub=Courier+Dover+Publications&rft.date=2007&rft.isbn=9780486458687&rft.aulast=Wussing&rft.aufirst=Hans&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DXp3JymnfAq4C%26pg%3DPA94&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+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">especially when the algebraic properties of the permutation are of interest.</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="CITEREFBiggsWhite1979" class="citation book cs1">Biggs, Norman L.; White, A. T. (1979). <i>Permutation groups and combinatorial structures</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-22287-7" title="Special:BookSources/0-521-22287-7"><bdi>0-521-22287-7</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+and+combinatorial+structures&rft.pub=Cambridge+University+Press&rft.date=1979&rft.isbn=0-521-22287-7&rft.aulast=Biggs&rft.aufirst=Norman+L.&rft.au=White%2C+A.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+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"><a href="#CITEREFRotman2006">Rotman 2006</a>, p. 107 – note especially the footnote on this page.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"> <a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 3 – see the comment following Example 1.2.2</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="CITEREFCameron1999" class="citation book cs1">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>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-65302-9" title="Special:BookSources/0-521-65302-9"><bdi>0-521-65302-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Permutation+groups&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=0-521-65302-9&rft.aulast=Cameron&rft.aufirst=Peter+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fpermutationgroup0000came&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+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="CITEREFJerrum1986" class="citation journal cs1">Jerrum, M. (1986). "A compact representation of permutation groups". <i>J. Algorithms</i>. <b>7</b> (1): 60–78. <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%2F0196-6774%2886%2990038-6">10.1016/0196-6774(86)90038-6</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Algorithms&rft.atitle=A+compact+representation+of+permutation+groups&rft.volume=7&rft.issue=1&rft.pages=60-78&rft.date=1986&rft_id=info%3Adoi%2F10.1016%2F0196-6774%2886%2990038-6&rft.aulast=Jerrum&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+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"><a href="#CITEREFRotman2006">Rotman 2006</a>, p. 108</span> </li> <li id="cite_note-Dixon96-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Dixon96_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Dixon96_12-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Dixon96_12-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 5</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFArtin1991">Artin 1991</a>, p. 177</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"><a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 17</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 18</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFCameron1994">Cameron 1994</a>, p. 228</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCameron1990" class="citation book cs1"><a href="/wiki/Peter_Cameron_(mathematician)" title="Peter Cameron (mathematician)">Cameron, Peter J.</a> (1990). <i>Oligomorphic permutation groups</i>. London Mathematical Society Lecture Note Series. Vol. 152. Cambridge: <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/0-521-38836-8" title="Special:BookSources/0-521-38836-8"><bdi>0-521-38836-8</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0813.20002">0813.20002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oligomorphic+permutation+groups&rft.place=Cambridge&rft.series=London+Mathematical+Society+Lecture+Note+Series&rft.pub=Cambridge+University+Press&rft.date=1990&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0813.20002%23id-name%3DZbl&rft.isbn=0-521-38836-8&rft.aulast=Cameron&rft.aufirst=Peter+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.newton.ac.uk/files/preprints/ni08029.pdf">Oligomorphic permutation groups</a> - Isaac Newton Institute preprint, Peter J. Cameron</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBhattacharjeeMacphersonMöllerNeumann1998" class="citation book cs1">Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.; Neumann, Peter M. (1998). <i>Notes on infinite permutation groups</i>. Lecture Notes in Mathematics. Vol. 1698. Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 83. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-64965-4" title="Special:BookSources/3-540-64965-4"><bdi>3-540-64965-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0916.20002">0916.20002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+infinite+permutation+groups&rft.place=Berlin&rft.series=Lecture+Notes+in+Mathematics&rft.pages=83&rft.pub=Springer-Verlag&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0916.20002%23id-name%3DZbl&rft.isbn=3-540-64965-4&rft.aulast=Bhattacharjee&rft.aufirst=Meenaxi&rft.au=Macpherson%2C+Dugald&rft.au=M%C3%B6ller%2C+R%C3%B6gnvaldur+G.&rft.au=Neumann%2C+Peter+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><a href="#CITEREFDixonMortimer1996">Dixon & Mortimer 1996</a>, p. 28</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><a href="#CITEREFCameron1994">Cameron 1994</a>, p. 226</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurnside1955" class="citation cs2">Burnside, William (1955) [1911], <i>Theory of Groups of Finite Order</i> (2nd ed.), Dover</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+Groups+of+Finite+Order&rft.edition=2nd&rft.pub=Dover&rft.date=1955&rft.aulast=Burnside&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWielandt1964" class="citation cs2">Wielandt, H. (1964), <i>Finite Permutation Groups</i>, Academic Press</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Finite+Permutation+Groups&rft.pub=Academic+Press&rft.date=1964&rft.aulast=Wielandt&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin1991" class="citation cs2"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (1991), <i>Algebra</i>, Prentice-Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-004763-5" title="Special:BookSources/0-13-004763-5"><bdi>0-13-004763-5</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.pub=Prentice-Hall&rft.date=1991&rft.isbn=0-13-004763-5&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCameron1994" class="citation cs2">Cameron, Peter J. (1994), <i>Combinatorics: Topics, Techniques, Algorithms</i>, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-45761-0" title="Special:BookSources/0-521-45761-0"><bdi>0-521-45761-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorics%3A+Topics%2C+Techniques%2C+Algorithms&rft.pub=Cambridge+University+Press&rft.date=1994&rft.isbn=0-521-45761-0&rft.aulast=Cameron&rft.aufirst=Peter+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+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 163), Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94599-7" title="Special:BookSources/0-387-94599-7"><bdi>0-387-94599-7</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=Graduate+Texts+in+Mathematics+163%29&rft.pub=Springer-Verlag&rft.date=1996&rft.isbn=0-387-94599-7&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%3APermutation+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman2006" class="citation cs2">Rotman, Joseph J. (2006), <i>A First Course in Abstract Algebra with Applications</i> (3rd ed.), Pearson Prentice-Hall, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-186267-7" title="Special:BookSources/0-13-186267-7"><bdi>0-13-186267-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Abstract+Algebra+with+Applications&rft.edition=3rd&rft.pub=Pearson+Prentice-Hall&rft.date=2006&rft.isbn=0-13-186267-7&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Permutation_group&action=edit&section=16" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Akos Seress. <i>Permutation group algorithms</i>. Cambridge Tracts in Mathematics, 152. Cambridge University Press, Cambridge, 2003.</li> <li>Meenaxi Bhattacharjee, Dugald Macpherson, Rögnvaldur G. Möller and Peter M. Neumann. <i>Notes on Infinite Permutation Groups</i>. Number 1698 in Lecture Notes in Mathematics. Springer-Verlag, 1998.</li> <li><a href="/wiki/Peter_Cameron_(mathematician)" title="Peter Cameron (mathematician)">Peter J. Cameron</a>. <i>Permutation Groups</i>. LMS Student Text 45. Cambridge University Press, Cambridge, 1999.</li> <li>Peter J. Cameron. <i>Oligomorphic Permutation Groups</i>. Cambridge University Press, Cambridge, 1990.</li></ul> <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=Permutation_group&action=edit&section=17" 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=Permutation_group">"Permutation 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=Permutation+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DPermutation_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3APermutation+group" class="Z3988"></span></li> <li>Alexander Hulpke. GAP Data Library <a rel="nofollow" class="external text" href="http://www.gap-system.org/Datalib/trans.html">"Transitive Permutation Groups"</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Permutation_group&oldid=1259395764">https://en.wikipedia.org/w/index.php?title=Permutation_group&oldid=1259395764</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:Finite_groups" title="Category:Finite groups">Finite 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_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</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 24 November 2024, at 22:43<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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Permutation group - Wikipedia