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<button aria-controls="toc-Main_classes_of_groups-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Main classes of groups subsection</span> </button> <ul id="toc-Main_classes_of_groups-sublist" class="vector-toc-list"> <li id="toc-Permutation_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Permutation_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Permutation groups</span> </div> </a> <ul id="toc-Permutation_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Matrix groups</span> </div> </a> <ul id="toc-Matrix_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Transformation_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transformation_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Transformation groups</span> </div> </a> <ul id="toc-Transformation_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abstract_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Abstract groups</span> </div> </a> <ul id="toc-Abstract_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Groups_with_additional_structure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Groups_with_additional_structure"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Groups with additional structure</span> </div> </a> <ul id="toc-Groups_with_additional_structure-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Branches_of_group_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Branches_of_group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Branches of group theory</span> </div> </a> <button aria-controls="toc-Branches_of_group_theory-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 Branches of group theory subsection</span> </button> <ul id="toc-Branches_of_group_theory-sublist" class="vector-toc-list"> <li id="toc-Finite_group_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Finite_group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Finite group theory</span> </div> </a> <ul id="toc-Finite_group_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation_of_groups" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representation_of_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Representation of groups</span> </div> </a> <ul id="toc-Representation_of_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lie_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lie_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Lie theory</span> </div> </a> <ul id="toc-Lie_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorial_and_geometric_group_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorial_and_geometric_group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Combinatorial and geometric group theory</span> </div> </a> <ul id="toc-Combinatorial_and_geometric_group_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Connection_of_groups_and_symmetry" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Connection_of_groups_and_symmetry"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Connection of groups and symmetry</span> </div> </a> <ul id="toc-Connection_of_groups_and_symmetry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications_of_group_theory" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications_of_group_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications of group theory</span> </div> </a> <button aria-controls="toc-Applications_of_group_theory-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 Applications of group theory subsection</span> </button> <ul id="toc-Applications_of_group_theory-sublist" class="vector-toc-list"> <li id="toc-Galois_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Galois_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Galois theory</span> </div> </a> <ul id="toc-Galois_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Algebraic topology</span> </div> </a> <ul id="toc-Algebraic_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_geometry" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Algebraic geometry</span> </div> </a> <ul id="toc-Algebraic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_number_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_number_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Algebraic number theory</span> </div> </a> <ul id="toc-Algebraic_number_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Harmonic_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Harmonic_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Harmonic analysis</span> </div> </a> <ul id="toc-Harmonic_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Combinatorics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Combinatorics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Combinatorics</span> </div> </a> <ul id="toc-Combinatorics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Music" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Music"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Music</span> </div> </a> <ul id="toc-Music-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Physics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Physics</span> </div> </a> <ul id="toc-Physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chemistry_and_materials_science" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Chemistry_and_materials_science"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Chemistry and materials science</span> </div> </a> <ul id="toc-Chemistry_and_materials_science-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cryptography" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cryptography"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.10</span> <span>Cryptography</span> </div> </a> <ul id="toc-Cryptography-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Group theory</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" 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Available in 73 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-73" 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">73 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%B2%D9%85%D8%B1" title="نظرية الزمر – Arabic" lang="ar" hreflang="ar" data-title="نظرية الزمر" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – Aragonese" lang="an" hreflang="an" data-title="Teoría de grupos" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – Asturian" lang="ast" hreflang="ast" data-title="Teoría de grupos" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qrup_n%C9%99z%C9%99riyy%C9%99si" title="Qrup nəzəriyyəsi – Azerbaijani" lang="az" hreflang="az" data-title="Qrup nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C3%BBn-l%C5%ABn" title="Kûn-lūn – Minnan" lang="nan" hreflang="nan" data-title="Kûn-lūn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D0%B4%D3%99%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Төркөмдәр теорияһы – Bashkir" lang="ba" hreflang="ba" data-title="Төркөмдәр теорияһы" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B3%D1%80%D1%83%D0%BF" title="Тэорыя груп – Belarusian" lang="be" hreflang="be" data-title="Тэорыя груп" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%B0%D1%9E" title="Тэорыя групаў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэорыя групаў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B8%D1%82%D0%B5" title="Теория на групите – Bulgarian" lang="bg" hreflang="bg" data-title="Теория на групите" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Teorija_grupa" title="Teorija grupa – Bosnian" lang="bs" hreflang="bs" data-title="Teorija grupa" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Damkaniezh_ar_strollo%C3%B9" title="Damkaniezh ar strolloù – Breton" lang="br" hreflang="br" data-title="Damkaniezh ar strolloù" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_grups" title="Teoria de grups – Catalan" lang="ca" hreflang="ca" data-title="Teoria de grups" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D1%88%D0%BA%C4%83%D0%BD%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Ушкăнсен теорийĕ – Chuvash" lang="cv" hreflang="cv" data-title="Ушкăнсен теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_grup" title="Teorie grup – Czech" lang="cs" hreflang="cs" data-title="Teorie grup" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Damcaniaeth_grwpiau" title="Damcaniaeth grwpiau – Welsh" lang="cy" hreflang="cy" data-title="Damcaniaeth grwpiau" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gruppeteori_(matematik)" title="Gruppeteori (matematik) – Danish" lang="da" hreflang="da" data-title="Gruppeteori (matematik)" 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/Gruppentheorie" title="Gruppentheorie – German" lang="de" hreflang="de" data-title="Gruppentheorie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%BF%CE%BC%CE%AC%CE%B4%CF%89%CE%BD" title="Θεωρία ομάδων – Greek" lang="el" hreflang="el" data-title="Θεωρία ομάδων" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – Spanish" lang="es" hreflang="es" data-title="Teoría de grupos" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grupo-teorio" title="Grupo-teorio – Esperanto" lang="eo" hreflang="eo" data-title="Grupo-teorio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde-teoria" title="Talde-teoria – Basque" lang="eu" hreflang="eu" data-title="Talde-teoria" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D9%88%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه گروه‌ها – Persian" lang="fa" hreflang="fa" data-title="نظریه گروه‌ها" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Th%C3%A9orie_des_groupes" title="Théorie des groupes – French" lang="fr" hreflang="fr" data-title="Théorie des groupes" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Teorie_dai_grups" title="Teorie dai grups – Friulian" lang="fur" hreflang="fur" data-title="Teorie dai grups" data-language-autonym="Furlan" data-language-local-name="Friulian" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – Galician" lang="gl" hreflang="gl" data-title="Teoría de grupos" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0%EB%A1%A0" title="군론 – Korean" lang="ko" hreflang="ko" data-title="군론" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%82%E0%A4%B9_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4" title="समूह सिद्धांत – Hindi" lang="hi" hreflang="hi" data-title="समूह सिद्धांत" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Teorija_grupa" title="Teorija grupa – Croatian" lang="hr" hreflang="hr" data-title="Teorija grupa" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Teori_grup" title="Teori grup – Indonesian" lang="id" hreflang="id" data-title="Teori grup" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Theoria_de_gruppos" title="Theoria de gruppos – Interlingua" lang="ia" hreflang="ia" data-title="Theoria de gruppos" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_dei_gruppi" title="Teoria dei gruppi – Italian" lang="it" hreflang="it" data-title="Teoria dei gruppi" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA" title="תורת החבורות – Hebrew" lang="he" hreflang="he" data-title="תורת החבורות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AF%E1%83%92%E1%83%A3%E1%83%A4%E1%83%97%E1%83%90_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="ჯგუფთა თეორია – Georgian" lang="ka" hreflang="ka" data-title="ჯგუფთა თეორია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Theoria_catervarum" title="Theoria catervarum – Latin" lang="la" hreflang="la" data-title="Theoria catervarum" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Grupu_teorija" title="Grupu teorija – Latvian" lang="lv" hreflang="lv" data-title="Grupu teorija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Csoportelm%C3%A9let" title="Csoportelmélet – Hungarian" lang="hu" hreflang="hu" data-title="Csoportelmélet" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82" title="ഗ്രൂപ്പ് സിദ്ധാന്തം – Malayalam" lang="ml" hreflang="ml" data-title="ഗ്രൂപ്പ് സിദ്ധാന്തം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kumpulan" title="Teori kumpulan – Malay" lang="ms" hreflang="ms" data-title="Teori kumpulan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Groepentheorie" title="Groepentheorie – Dutch" lang="nl" hreflang="nl" data-title="Groepentheorie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%AA_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="ग्रुप सिद्धान्त – Newari" lang="new" hreflang="new" data-title="ग्रुप सिद्धान्त" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%BE%A4%E8%AB%96" title="群論 – Japanese" lang="ja" hreflang="ja" data-title="群論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gruppeteori" title="Gruppeteori – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Gruppeteori" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Gruppeteori" title="Gruppeteori – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Gruppeteori" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%B0%E0%A9%81%E0%A9%B1%E0%A8%AA_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A9%80" title="ਗਰੁੱਪ ਥਿਊਰੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਗਰੁੱਪ ਥਿਊਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%B9%D9%88%D9%84%DB%8C_%D8%B3%D9%88%DA%86" title="ٹولی سوچ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ٹولی سوچ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_grup" title="Teoria grup – Polish" lang="pl" hreflang="pl" data-title="Teoria grup" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Teoria_dos_grupos" title="Teoria dos grupos – Portuguese" lang="pt" hreflang="pt" data-title="Teoria dos grupos" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_grupurilor" title="Teoria grupurilor – Romanian" lang="ro" hreflang="ro" data-title="Teoria grupurilor" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D2%91%D1%80%D1%83%D0%BF" title="Теорія ґруп – Rusyn" lang="rue" hreflang="rue" data-title="Теорія ґруп" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF" title="Теория групп – Russian" lang="ru" hreflang="ru" data-title="Теория групп" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Group_theory" title="Group theory – Scots" lang="sco" hreflang="sco" data-title="Group theory" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Teoria_e_grupeve" title="Teoria e grupeve – Albanian" lang="sq" hreflang="sq" data-title="Teoria e grupeve" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Group_theory" title="Group theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Group theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Te%C3%B3ria_gr%C3%BAp" title="Teória grúp – Slovak" lang="sk" hreflang="sk" data-title="Teória grúp" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Teorija_grup" title="Teorija grup – Slovenian" lang="sl" hreflang="sl" data-title="Teorija grup" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%DA%AF%D8%B1%D9%88%D9%88%D9%BE" title="تیۆریی گرووپ – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تیۆریی گرووپ" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Теорија група – Serbian" lang="sr" hreflang="sr" data-title="Теорија група" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_grupa" title="Teorija grupa – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Teorija grupa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ryhm%C3%A4teoria" title="Ryhmäteoria – Finnish" lang="fi" hreflang="fi" data-title="Ryhmäteoria" 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/Gruppteori" title="Gruppteori – Swedish" lang="sv" hreflang="sv" data-title="Gruppteori" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Teorya_ng_grupo" title="Teorya ng grupo – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng grupo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="குலக் கோட்பாடு – Tamil" lang="ta" hreflang="ta" data-title="குலக் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%97%E0%B8%A4%E0%B8%A9%E0%B8%8E%E0%B8%B5%E0%B8%81%E0%B8%A3%E0%B8%B8%E0%B8%9B" title="ทฤษฎีกรุป – Thai" lang="th" hreflang="th" data-title="ทฤษฎีกรุป" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Grup_teorisi" title="Grup teorisi – Turkish" lang="tr" hreflang="tr" data-title="Grup teorisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B3%D1%80%D1%83%D0%BF" title="Теорія груп – Ukrainian" lang="uk" hreflang="uk" data-title="Теорія груп" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%DA%AF%D8%B1%D9%88%DB%81" title="نظریۂ گروہ – Urdu" lang="ur" hreflang="ur" data-title="نظریۂ 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/L%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m" title="Lý thuyết nhóm – Vietnamese" lang="vi" hreflang="vi" data-title="Lý thuyết nhóm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Teyorya_grupo" title="Teyorya grupo – Waray" lang="war" hreflang="war" data-title="Teyorya grupo" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BE%A4%E8%AE%BA" title="群论 – Wu" lang="wuu" hreflang="wuu" data-title="群论" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%A8%D7%95%D7%A4%D7%A2_%D7%98%D7%A2%D7%90%D7%A8%D7%99%D7%A2" title="גרופע טעאריע – Yiddish" lang="yi" hreflang="yi" data-title="גרופע טעאריע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%BE%A3%E8%AB%96" title="羣論 – Cantonese" lang="yue" hreflang="yue" data-title="羣論" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Gropiu_teuor%C4%97j%C4%97" title="Gropiu teuorėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Gropiu teuorėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BE%A4%E8%AE%BA" title="群论 – Chinese" lang="zh" 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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 class="mw-selflink selflink">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> <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 <a href="/wiki/Rubik%27s_Cube" title="Rubik's Cube">Rubik's Cube</a> puzzle, invented in 1974 by <a href="/wiki/Ern%C5%91_Rubik" title="Ernő Rubik">Ernő Rubik</a>, has been used as an illustration of <a href="/wiki/Permutation_group" title="Permutation group">permutation groups</a>. See <a href="/wiki/Rubik%27s_Cube_group" title="Rubik's Cube group">Rubik's Cube group</a>.</figcaption></figure> <p>In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, <b>group theory</b> studies the <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> known as <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a>. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>, and <a href="/wiki/Vector_space" title="Vector space">vector spaces</a>, can all be seen as groups endowed with additional <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> and <a href="/wiki/Axiom" title="Axiom">axioms</a>. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. <a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic groups</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> are two branches of group theory that have experienced advances and have become subject areas in their own right. </p><p>Various physical systems, such as <a href="/wiki/Crystal" title="Crystal">crystals</a> and the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a>, and <a href="/wiki/Standard_Model" title="Standard Model">three of the four</a> known fundamental forces in the universe, may be modelled by <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry groups</a>. Thus group theory and the closely related <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> have many important applications in <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, and <a href="/wiki/Materials_science" title="Materials science">materials science</a>. Group theory is also central to <a href="/wiki/Public_key_cryptography" class="mw-redirect" title="Public key cryptography">public key cryptography</a>. </p><p>The early <a href="/wiki/History_of_group_theory" title="History of group theory">history of group theory</a> dates from the 19th century. One of the most important mathematical achievements of the 20th century<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> was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=1" 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>Group theory has three main historical sources: <a href="/wiki/Number_theory" title="Number theory">number theory</a>, the theory of <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equations</a>, and <a href="/wiki/Geometry" title="Geometry">geometry</a>. The number-theoretic strand was begun by <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>, and developed by <a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss's</a> work on <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> and additive and multiplicative groups related to <a href="/wiki/Quadratic_field" title="Quadratic field">quadratic fields</a>. Early results about permutation groups were obtained by <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a>, <a href="/wiki/Paolo_Ruffini_(mathematician)" class="mw-redirect" title="Paolo Ruffini (mathematician)">Ruffini</a>, and <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Abel</a> in their quest for general solutions of polynomial equations of high degree. <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> coined the term "group" and established a connection, now known as <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a>, between the nascent theory of groups and <a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">field theory</a>. In geometry, groups first became important in <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a> and, later, <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a>. <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>'s <a href="/wiki/Erlangen_program" title="Erlangen program">Erlangen program</a> proclaimed group theory to be the organizing principle of geometry. </p><p><a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Galois</a>, in the 1830s, was the first to employ groups to determine the solvability of <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a>. <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a> and <a href="/wiki/Augustin_Louis_Cauchy" class="mw-redirect" title="Augustin Louis Cauchy">Augustin Louis Cauchy</a> pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from <a href="/wiki/Geometry" title="Geometry">geometrical</a> situations. In an attempt to come to grips with possible geometries (such as <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">euclidean</a>, <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic</a> or <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>) using group theory, <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> initiated the <a href="/wiki/Erlangen_programme" class="mw-redirect" title="Erlangen programme">Erlangen programme</a>. <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a>, in 1884, started using groups (now called <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>) attached to <a href="/wiki/Analysis_(mathematics)" class="mw-redirect" title="Analysis (mathematics)">analytic</a> problems. Thirdly, groups were, at first implicitly and later explicitly, used in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. </p><p>The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> in the early 20th century, <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, and many more influential spin-off domains. The <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> is a vast body of work from the mid 20th century, classifying all the <a href="/wiki/Finite_set" title="Finite set">finite</a> <a href="/wiki/Simple_group" title="Simple group">simple groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Main_classes_of_groups">Main classes of groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=2" title="Edit section: Main classes of groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group (mathematics)</a></div> <p>The range of groups being considered has gradually expanded from finite permutation groups and special examples of <a href="/wiki/Matrix_group" class="mw-redirect" title="Matrix group">matrix groups</a> to abstract groups that may be specified through a <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> by <a href="/wiki/Generating_set_of_a_group" title="Generating set of a group">generators</a> and <a href="/wiki/Binary_relation" title="Binary relation">relations</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Permutation_groups">Permutation groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=3" title="Edit section: Permutation groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first <a href="/wiki/Class_(set_theory)" title="Class (set theory)">class</a> of groups to undergo a systematic study was <a href="/wiki/Permutation_group" title="Permutation group">permutation groups</a>. Given any set <i>X</i> and a collection <i>G</i> of <a href="/wiki/Bijection" title="Bijection">bijections</a> of <i>X</i> into itself (known as <i>permutations</i>) that is closed under compositions and inverses, <i>G</i> is a group <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acting</a> on <i>X</i>. If <i>X</i> consists of <i>n</i> elements and <i>G</i> consists of <i>all</i> permutations, <i>G</i> is the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> S<sub><i>n</i></sub>; in general, any permutation group <i>G</i> is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the symmetric group of <i>X</i>. An early construction due to <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Cayley</a> exhibited any group as a permutation group, acting on itself (<span class="nowrap"><i>X</i> = <i>G</i></span>) by means of the left <a href="/wiki/Regular_representation" title="Regular representation">regular representation</a>. </p><p>In many cases, the structure of a permutation group can be studied using the properties of its action on the corresponding set. For example, in this way one proves that for <span class="nowrap"><i>n</i> ≥ 5</span>, the <a href="/wiki/Alternating_group" title="Alternating group">alternating group</a> A<sub><i>n</i></sub> is <a href="/wiki/Simple_group" title="Simple group">simple</a>, i.e. does not admit any proper <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroups</a>. This fact plays a key role in the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">impossibility of solving a general algebraic equation of degree <span class="nowrap"><i>n</i> ≥ 5</span> in radicals</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_groups">Matrix groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=4" title="Edit section: Matrix groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The next important class of groups is given by <i>matrix groups</i>, or <a href="/wiki/Linear_group" title="Linear group">linear groups</a>. Here <i>G</i> is a set consisting of invertible <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> of given order <i>n</i> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i> that is closed under the products and inverses. Such a group acts on the <i>n</i>-dimensional vector space <i>K</i><sup><i>n</i></sup> by <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a>. This action makes matrix groups conceptually similar to permutation groups, and the geometry of the action may be usefully exploited to establish properties of the group <i>G</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Transformation_groups">Transformation groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=5" title="Edit section: Transformation groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Permutation groups and matrix groups are special cases of <a href="/wiki/Transformation_group" class="mw-redirect" title="Transformation group">transformation groups</a>: groups that act on a certain space <i>X</i> preserving its inherent structure. In the case of permutation groups, <i>X</i> is a set; for matrix groups, <i>X</i> is a <a href="/wiki/Vector_space" title="Vector space">vector space</a>. The concept of a transformation group is closely related with the concept of a <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a>: transformation groups frequently consist of <i>all</i> transformations that preserve a certain structure. </p><p>The theory of transformation groups forms a bridge connecting group theory with <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a>. A long line of research, originating with <a href="/wiki/Sophus_Lie" title="Sophus Lie">Lie</a> and <a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a>, considers group actions on <a href="/wiki/Manifold" title="Manifold">manifolds</a> by <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphisms</a> or <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphisms</a>. The groups themselves may be <a href="/wiki/Discrete_group" title="Discrete group">discrete</a> or <a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Abstract_groups">Abstract groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=6" title="Edit section: Abstract groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most groups considered in the first stage of the development of group theory were "concrete", having been realized through numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an <b>abstract group</b> began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two <a href="/wiki/Group_isomorphism" title="Group isomorphism">isomorphic groups</a> are considered as the same group. A typical way of specifying an abstract group is through a <a href="/wiki/Presentation_of_a_group" title="Presentation of a group">presentation</a> by <i>generators and relations</i>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\langle S|R\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>R</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\langle S|R\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76de52f6daa00e539c8cdb3cb1847437e2ed664b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.291ex; height:2.843ex;" alt="{\displaystyle G=\langle S|R\rangle .}"></span></dd></dl> <p>A significant source of abstract groups is given by the construction of a <i>factor group</i>, or <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a>, <i>G</i>/<i>H</i>, of a group <i>G</i> by a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> <i>H</i>. <a href="/wiki/Class_group" class="mw-redirect" title="Class group">Class groups</a> of <a href="/wiki/Algebraic_number_field" title="Algebraic number field">algebraic number fields</a> were among the earliest examples of factor groups, of much interest in <a href="/wiki/Number_theory" title="Number theory">number theory</a>. If a group <i>G</i> is a permutation group on a set <i>X</i>, the factor group <i>G</i>/<i>H</i> is no longer acting on <i>X</i>; but the idea of an abstract group permits one not to worry about this discrepancy. </p><p>The change of perspective from concrete to abstract groups makes it natural to consider properties of groups that are independent of a particular realization, or in modern language, invariant under <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a>, as well as the classes of group with a given such property: <a href="/wiki/Finite_group" title="Finite group">finite groups</a>, <a href="/wiki/Periodic_group" class="mw-redirect" title="Periodic group">periodic groups</a>, <a href="/wiki/Simple_group" title="Simple group">simple groups</a>, <a href="/wiki/Solvable_group" title="Solvable group">solvable groups</a>, and so on. Rather than exploring properties of an individual group, one seeks to establish results that apply to a whole class of groups. The new paradigm was of paramount importance for the development of mathematics: it foreshadowed the creation of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> in the works of <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, <a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a>, <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>, and mathematicians of their school.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (June 2012)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Groups_with_additional_structure">Groups with additional structure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=7" title="Edit section: Groups with additional structure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important elaboration of the concept of a group occurs if <i>G</i> is endowed with additional structure, notably, of a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, or <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>. If the group operations <i>m</i> (multiplication) and <i>i</i> (inversion), </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 m:G\times G\to G,(g,h)\mapsto gh,\quad i:G\to G,g\mapsto g^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>:</mo> <mi>G</mi> <mo>×</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>,</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>g</mi> <mi>h</mi> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">↦</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m:G\times G\to G,(g,h)\mapsto gh,\quad i:G\to G,g\mapsto g^{-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f405c2807a694268add30e4c1aa4f43cb0df05e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.539ex; height:3.176ex;" alt="{\displaystyle m:G\times G\to G,(g,h)\mapsto gh,\quad i:G\to G,g\mapsto g^{-1},}"></span></dd></dl> <p>are compatible with this structure, that is, they are <a href="/wiki/Continuous_map" class="mw-redirect" title="Continuous map">continuous</a>, <a href="/wiki/Smooth_map" class="mw-redirect" title="Smooth map">smooth</a> or <a href="/wiki/Regular_map_(algebraic_geometry)" class="mw-redirect" title="Regular map (algebraic geometry)">regular</a> (in the sense of algebraic geometry) maps, then <i>G</i> is a <a href="/wiki/Topological_group" title="Topological group">topological group</a>, a <a href="/wiki/Lie_group" title="Lie group">Lie group</a>, or an <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic group</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The presence of extra structure relates these types of groups with other mathematical disciplines and means that more tools are available in their study. Topological groups form a natural domain for <a href="/wiki/Abstract_harmonic_analysis" class="mw-redirect" title="Abstract harmonic analysis">abstract harmonic analysis</a>, whereas <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> (frequently realized as transformation groups) are the mainstays of <a href="/wiki/Differential_geometry" title="Differential geometry">differential geometry</a> and unitary <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>. Certain classification questions that cannot be solved in general can be approached and resolved for special subclasses of groups. Thus, <a href="/wiki/Compact_Lie_group" class="mw-redirect" title="Compact Lie group">compact connected Lie groups</a> have been completely classified. There is a fruitful relation between infinite abstract groups and topological groups: whenever a group <i>Γ</i> can be realized as a <a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">lattice</a> in a topological group <i>G</i>, the geometry and analysis pertaining to <i>G</i> yield important results about <i>Γ</i>. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (<a href="/wiki/Profinite_group" title="Profinite group">profinite groups</a>): for example, a single <a href="/wiki/Powerful_p-group" title="Powerful p-group"><i>p</i>-adic analytic group</a> <i>G</i> has a family of quotients which are finite <a href="/wiki/P-group" title="P-group"><i>p</i>-groups</a> of various orders, and properties of <i>G</i> translate into the properties of its finite quotients. </p> <div class="mw-heading mw-heading2"><h2 id="Branches_of_group_theory">Branches of group theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=8" title="Edit section: Branches of group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Finite_group_theory">Finite group theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=9" title="Edit section: Finite group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Finite_group" title="Finite group">Finite group</a></div> <p>During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the <a href="/wiki/Local_analysis" title="Local analysis">local theory</a> of finite groups and the theory of <a href="/wiki/Solvable_group" title="Solvable group">solvable</a> and <a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent groups</a>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="In who's opinion? (December 2013)">citation needed</span></a></i>]</sup> As a consequence, the complete <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a> was achieved, meaning that all those <a href="/wiki/Simple_group" title="Simple group">simple groups</a> from which all finite groups can be built are now known. </p><p>During the second half of the twentieth century, mathematicians such as <a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Chevalley</a> and <a href="/wiki/Robert_Steinberg" title="Robert Steinberg">Steinberg</a> also increased our understanding of finite analogs of <a href="/wiki/Classical_group" title="Classical group">classical groups</a>, and other related groups. One such family of groups is the family of <a href="/wiki/General_linear_group" title="General linear group">general linear groups</a> over <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. Finite groups often occur when considering <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> of mathematical or physical objects, when those objects admit just a finite number of structure-preserving transformations. The theory of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a>, which may be viewed as dealing with "<a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetry</a>", is strongly influenced by the associated <a href="/wiki/Weyl_group" title="Weyl group">Weyl groups</a>. These are finite groups generated by reflections which act on a finite-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. The properties of finite groups can thus play a role in subjects such as <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Representation_of_groups">Representation of groups</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=10" title="Edit section: Representation of groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></div> <p>Saying that a group <i>G</i> <i><a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">acts</a></i> on a set <i>X</i> means that every element of <i>G</i> defines a bijective map on the set <i>X</i> in a way compatible with the group structure. When <i>X</i> has more structure, it is useful to restrict this notion further: a representation of <i>G</i> on a <a href="/wiki/Vector_space" title="Vector space">vector space</a> <i>V</i> is a <a href="/wiki/Group_homomorphism" title="Group homomorphism">group homomorphism</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho :G\to \operatorname {GL} (V),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho :G\to \operatorname {GL} (V),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837c87a8f0b89992163537a4542797765324bf1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.1ex; height:2.843ex;" alt="{\displaystyle \rho :G\to \operatorname {GL} (V),}"></span></dd></dl> <p>where <a href="/wiki/General_linear_group" title="General linear group">GL</a>(<i>V</i>) consists of the invertible <a href="/wiki/Linear_map" title="Linear map">linear transformations</a> of <i>V</i>. In other words, to every group element <i>g</i> is assigned an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> <i>ρ</i>(<i>g</i>) such that <span class="nowrap"><i>ρ</i>(<i>g</i>) ∘ <i>ρ</i>(<i>h</i>) = <i>ρ</i>(<i>gh</i>)</span> for any <i>h</i> in <i>G</i>. </p><p>This definition can be understood in two directions, both of which give rise to whole new domains of mathematics.<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> On the one hand, it may yield new information about the group <i>G</i>: often, the group operation in <i>G</i> is abstractly given, but via <i>ρ</i>, it corresponds to the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication of matrices</a>, which is very explicit.<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> On the other hand, given a well-understood group acting on a complicated object, this simplifies the study of the object in question. For example, if <i>G</i> is finite, it is known that <i>V</i> above decomposes into <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible parts</a> (see <a href="/wiki/Maschke%27s_theorem" title="Maschke's theorem">Maschke's theorem</a>). These parts, in turn, are much more easily manageable than the whole <i>V</i> (via <a href="/wiki/Schur%27s_lemma" title="Schur's lemma">Schur's lemma</a>). </p><p>Given a group <i>G</i>, <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> then asks what representations of <i>G</i> exist. There are several settings, and the employed methods and obtained results are rather different in every case: <a href="/wiki/Representation_theory_of_finite_groups" title="Representation theory of finite groups">representation theory of finite groups</a> and representations of <a href="/wiki/Lie_group" title="Lie group">Lie groups</a> are two main subdomains of the theory. The totality of representations is governed by the group's <a href="/wiki/Character_theory" title="Character theory">characters</a>. For example, <a href="/wiki/Fourier_series" title="Fourier series">Fourier polynomials</a> can be interpreted as the characters of <a href="/wiki/Unitary_group" title="Unitary group">U(1)</a>, the group of <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> of <a href="/wiki/Absolute_value" title="Absolute value">absolute value</a> <i>1</i>, acting on the <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>2</sup></a>-space of periodic functions. </p> <div class="mw-heading mw-heading3"><h3 id="Lie_theory">Lie theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=11" title="Edit section: Lie theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a></div> <p>A <a href="/wiki/Lie_group" title="Lie group">Lie group</a> is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> that is also a <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, with the property that the group operations are compatible with the <a href="/wiki/Differential_structure" title="Differential structure">smooth structure</a>. Lie groups are named after <a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a>, who laid the foundations of the theory of continuous <a href="/wiki/Transformation_group" class="mw-redirect" title="Transformation group">transformation groups</a>. The term <i>groupes de Lie</i> first appeared in French in 1893 in the thesis of Lie's student <a href="/w/index.php?title=Arthur_Tresse&action=edit&redlink=1" class="new" title="Arthur Tresse (page does not exist)">Arthur Tresse</a>, page 3.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>Lie groups represent the best-developed theory of <a href="/wiki/Continuous_symmetry" title="Continuous symmetry">continuous symmetry</a> of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> and <a href="/wiki/Mathematical_structure" title="Mathematical structure">structures</a>, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a>. They provide a natural framework for analysing the continuous symmetries of <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a> (<a href="/wiki/Differential_Galois_theory" title="Differential Galois theory">differential Galois theory</a>), in much the same way as permutation groups are used in <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> for analysing the discrete symmetries of <a href="/wiki/Algebraic_equations" class="mw-redirect" title="Algebraic equations">algebraic equations</a>. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations. </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorial_and_geometric_group_theory">Combinatorial and geometric group theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=12" title="Edit section: Combinatorial and geometric group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Geometric_group_theory" title="Geometric group theory">Geometric group theory</a></div> <p>Groups can be described in different ways. Finite groups can be described by writing down the <a href="/wiki/Group_table" class="mw-redirect" title="Group table">group table</a> consisting of all possible multiplications <span class="nowrap"><i>g</i> • <i>h</i></span>. A more compact way of defining a group is by <i>generators and relations</i>, also called the <i>presentation</i> of a group. Given any set <i>F</i> of generators <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{g_{i}\}_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{g_{i}\}_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94097cd1abbfd1b93587e0cadd2833bde488afa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.958ex; height:2.843ex;" alt="{\displaystyle \{g_{i}\}_{i\in I}}"></span>, the <a href="/wiki/Free_group" title="Free group">free group</a> generated by <i>F</i> surjects onto the group <i>G</i>. The kernel of this map is called the subgroup of relations, generated by some subset <i>D</i>. The presentation is usually denoted 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 \langle F\mid D\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>F</mi> <mo>∣</mo> <mi>D</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle F\mid D\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d584b9fdb8b434f9ed7a35cb4e962028e0ac9756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.058ex; height:2.843ex;" alt="{\displaystyle \langle F\mid D\rangle .}"></span> For example, the group presentation <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 \langle a,b\mid aba^{-1}b^{-1}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∣</mo> <mi>a</mi> <mi>b</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle a,b\mid aba^{-1}b^{-1}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffb9d796a04a4f925d2a7e606f306baf27c7757f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.128ex; height:3.176ex;" alt="{\displaystyle \langle a,b\mid aba^{-1}b^{-1}\rangle }"></span> describes a group which is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \times \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \times \mathbb {Z} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad96c9aa929bbe178f58ed6efbf98093d4f73906" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.588ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \times \mathbb {Z} .}"></span> A string consisting of generator symbols and their inverses is called a <i>word</i>. </p><p><a href="/wiki/Combinatorial_group_theory" title="Combinatorial group theory">Combinatorial group theory</a> studies groups from the perspective of generators and relations.<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> It is particularly useful where finiteness assumptions are satisfied, for example finitely generated groups, or finitely presented groups (i.e. in addition the relations are finite). The area makes use of the connection of <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a> via their <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental groups</a>. A fundamental theorem of this area is that every subgroup of a free group is free. </p><p>There are several natural questions arising from giving a group by its presentation. The <i><a href="/wiki/Word_problem_for_groups" title="Word problem for groups">word problem</a></i> asks whether two words are effectively the same group element. By relating the problem to <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a>, one can show that there is in general no <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> solving this task. Another, generally harder, algorithmically insoluble problem is the <a href="/wiki/Group_isomorphism_problem" title="Group isomorphism problem">group isomorphism problem</a>, which asks whether two groups given by different presentations are actually isomorphic. For example, the group with presentation <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 \langle x,y\mid xyxyx=e\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∣</mo> <mi>x</mi> <mi>y</mi> <mi>x</mi> <mi>y</mi> <mi>x</mi> <mo>=</mo> <mi>e</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle x,y\mid xyxyx=e\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163336c1cfc547d82e70c87498435d225342b686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.394ex; height:2.843ex;" alt="{\displaystyle \langle x,y\mid xyxyx=e\rangle ,}"></span> is isomorphic to the additive group <b>Z</b> of integers, although this may not be immediately apparent. (Writing <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 z=xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46602cba442d7bb02978c0c8c328e180738e7d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.672ex; height:2.009ex;" alt="{\displaystyle z=xy}"></span>, one has <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cong \langle z,y\mid z^{3}=y\rangle \cong \langle z\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≅</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>∣</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>≅</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>z</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cong \langle z,y\mid z^{3}=y\rangle \cong \langle z\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eefae04532cb010b7d454046d20f6275cfb174b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.99ex; height:3.176ex;" alt="{\displaystyle G\cong \langle z,y\mid z^{3}=y\rangle \cong \langle z\rangle .}"></span>) </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Cayley_graph_of_F2.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cayley_graph_of_F2.svg/150px-Cayley_graph_of_F2.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cayley_graph_of_F2.svg/225px-Cayley_graph_of_F2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Cayley_graph_of_F2.svg/300px-Cayley_graph_of_F2.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>The Cayley graph of ⟨ x, y ∣ ⟩, the free group of rank 2</figcaption></figure> <p><a href="/wiki/Geometric_group_theory" title="Geometric group theory">Geometric group theory</a> attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on.<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> The first idea is made precise by means of the <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a>, whose vertices correspond to group elements and edges correspond to right multiplication in the group. Given two elements, one constructs the <a href="/wiki/Word_metric" title="Word metric">word metric</a> given by the length of the minimal path between the elements. A theorem of <a href="/wiki/John_Milnor" title="John Milnor">Milnor</a> and Svarc then says that given a group <i>G</i> acting in a reasonable manner on a <a href="/wiki/Metric_space" title="Metric space">metric space</a> <i>X</i>, for example a <a href="/wiki/Compact_manifold" class="mw-redirect" title="Compact manifold">compact manifold</a>, then <i>G</i> is <a href="/wiki/Quasi-isometry" title="Quasi-isometry">quasi-isometric</a> (i.e. looks similar from a distance) to the space <i>X</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Connection_of_groups_and_symmetry">Connection of groups and symmetry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=13" title="Edit section: Connection of groups and symmetry"><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/Symmetry_group" title="Symmetry group">Symmetry group</a></div> <p>Given a structured object <i>X</i> of any sort, a <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> is a mapping of the object onto itself which preserves the structure. This occurs in many cases, for example </p> <ul><li>If <i>X</i> is a set with no additional structure, a symmetry is a <a href="/wiki/Bijection" title="Bijection">bijective</a> map from the set to itself, giving rise to permutation groups.</li> <li>If the object <i>X</i> is a set of points in the plane with its <a href="/wiki/Metric_space" title="Metric space">metric</a> structure or any other <a href="/wiki/Metric_space" title="Metric space">metric space</a>, a symmetry is a <a href="/wiki/Bijection" title="Bijection">bijection</a> of the set to itself which preserves the distance between each pair of points (an <a href="/wiki/Isometry" title="Isometry">isometry</a>). The corresponding group is called <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a> of <i>X</i>.</li> <li>If instead <a href="/wiki/Angle" title="Angle">angles</a> are preserved, one speaks of <a href="/wiki/Conformal_map" title="Conformal map">conformal maps</a>. Conformal maps give rise to <a href="/wiki/Kleinian_group" title="Kleinian group">Kleinian groups</a>, for example.</li> <li>Symmetries are not restricted to geometrical objects, but include algebraic objects as well. For instance, the equation <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^{2}-3=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-3=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8549f2d0e76e6d61a6e1f32019e8ef8ff042eaf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle x^{2}-3=0}"></span> has the two solutions <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 {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b19c09494138b5082459afac7f9a8d99c546fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:2.843ex;" alt="{\displaystyle {\sqrt {3}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c7cc8150f4cf1328d10ea1883d66f073136647b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.906ex; height:2.843ex;" alt="{\displaystyle -{\sqrt {3}}}"></span>. In this case, the group that exchanges the two roots is the <a href="/wiki/Galois_group" title="Galois group">Galois group</a> belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots.</li></ul> <p>The axioms of a group formalize the essential aspects of <a href="/wiki/Symmetry" title="Symmetry">symmetry</a>. Symmetries form a group: they are <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> because if you take a symmetry of an object, and then apply another symmetry, the result will still be a symmetry. The identity keeping the object fixed is always a symmetry of an object. Existence of inverses is guaranteed by undoing the symmetry and the associativity comes from the fact that symmetries are functions on a space, and composition of functions is associative. </p><p><a href="/wiki/Frucht%27s_theorem" title="Frucht's theorem">Frucht's theorem</a> says that every group is the symmetry group of some <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>. So every abstract group is actually the symmetries of some explicit object. </p><p>The saying of "preserving the structure" of an object can be made precise by working in a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. Maps preserving the structure are then the <a href="/wiki/Morphism" title="Morphism">morphisms</a>, and the symmetry group is the <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of the object in question. </p> <div class="mw-heading mw-heading2"><h2 id="Applications_of_group_theory">Applications of group theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=14" title="Edit section: Applications of group theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Applications of group theory abound. Almost all structures in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> are special cases of groups. <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a>, for example, can be viewed as <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a> (corresponding to addition) together with a second operation (corresponding to multiplication). Therefore, group theoretic arguments underlie large parts of the theory of those entities. </p> <div class="mw-heading mw-heading3"><h3 id="Galois_theory">Galois theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=15" title="Edit section: Galois theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a></div> <p><a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> uses groups to describe the symmetries of the roots of a polynomial (or more precisely the automorphisms of the algebras generated by these roots). The <a href="/wiki/Fundamental_theorem_of_Galois_theory" title="Fundamental theorem of Galois theory">fundamental theorem of Galois theory</a> provides a link between <a href="/wiki/Algebraic_field_extension" class="mw-redirect" title="Algebraic field extension">algebraic field extensions</a> and group theory. It gives an effective criterion for the solvability of polynomial equations in terms of the solvability of the corresponding <a href="/wiki/Galois_group" title="Galois group">Galois group</a>. For example, <i>S</i><sub>5</sub>, the <a href="/wiki/Symmetric_group" title="Symmetric group">symmetric group</a> in 5 elements, is not solvable which implies that the general <a href="/wiki/Quintic_equation" class="mw-redirect" title="Quintic equation">quintic equation</a> cannot be solved by radicals in the way equations of lower degree can. The theory, being one of the historical roots of group theory, is still fruitfully applied to yield new results in areas such as <a href="/wiki/Class_field_theory" title="Class field theory">class field theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_topology">Algebraic topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=16" title="Edit section: Algebraic topology"><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/Algebraic_topology" title="Algebraic topology">Algebraic topology</a></div> <p><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic topology</a> is another domain which prominently <a href="/wiki/Functor" title="Functor">associates</a> groups to the objects the theory is interested in. There, groups are used to describe certain invariants of <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some <a href="/wiki/Homeomorphism" title="Homeomorphism">deformation</a>. For example, the <a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a> "counts" how many paths in the space are essentially different. The <a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a>, proved in 2002/2003 by <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a>, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of <a href="/wiki/Eilenberg%E2%80%93MacLane_space" title="Eilenberg–MacLane space">Eilenberg–MacLane spaces</a> which are spaces with prescribed <a href="/wiki/Homotopy_groups" class="mw-redirect" title="Homotopy groups">homotopy groups</a>. Similarly <a href="/wiki/Algebraic_K-theory" title="Algebraic K-theory">algebraic K-theory</a> relies in a way on <a href="/wiki/Classifying_space" title="Classifying space">classifying spaces</a> of groups. Finally, the name of the <a href="/wiki/Torsion_subgroup" title="Torsion subgroup">torsion subgroup</a> of an infinite group shows the legacy of topology in group theory. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Torus.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/200px-Torus.png" decoding="async" width="200" height="128" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/300px-Torus.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Torus.png/400px-Torus.png 2x" data-file-width="784" data-file-height="502" /></a><figcaption>A torus. Its abelian group structure is induced from the map <span class="nowrap"><b>C</b> → <b>C</b>/(<b>Z</b> + <i>τ</i><b>Z</b>)</span>, where <i>τ</i> is a parameter living in the <a href="/wiki/Upper_half_plane" class="mw-redirect" title="Upper half plane">upper half plane</a>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Algebraic_geometry">Algebraic geometry</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=17" title="Edit section: Algebraic geometry"><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/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a></div> <p><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic geometry</a> likewise uses group theory in many ways. <a href="/wiki/Abelian_variety" title="Abelian variety">Abelian varieties</a> have been introduced above. The presence of the group operation yields additional information which makes these varieties particularly accessible. They also often serve as a test for new conjectures. (For example the <a href="/wiki/Hodge_conjecture" title="Hodge conjecture">Hodge conjecture</a> (in certain cases).) The one-dimensional case, namely <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a> is studied in particular detail. They are both theoretically and practically intriguing.<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> In another direction, <a href="/wiki/Toric_variety" title="Toric variety">toric varieties</a> are <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> acted on by a <a href="/wiki/Torus" title="Torus">torus</a>. Toroidal embeddings have recently led to advances in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, in particular <a href="/wiki/Resolution_of_singularities" title="Resolution of singularities">resolution of singularities</a>.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Algebraic_number_theory">Algebraic number theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=18" title="Edit section: Algebraic number theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></div> <p><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> makes uses of groups for some important applications. For example, <a href="/wiki/Euler_product" title="Euler product">Euler's product formula</a>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {1}{n^{s}}}&=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},\\\end{aligned}}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msup> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {1}{n^{s}}}&=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},\\\end{aligned}}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ef08c8ae41db278dd51c94922fa03060d8fd9fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:25.286ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {1}{n^{s}}}&=\prod _{p{\text{ prime}}}{\frac {1}{1-p^{-s}}},\\\end{aligned}}\!}"></span></dd></dl> <p>captures <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">the fact</a> that any integer decomposes in a unique way into <a href="/wiki/Prime_number" title="Prime number">primes</a>. The failure of this statement for <a href="/wiki/Dedekind_ring" class="mw-redirect" title="Dedekind ring">more general rings</a> gives rise to <a href="/wiki/Class_group" class="mw-redirect" title="Class group">class groups</a> and <a href="/wiki/Regular_prime" title="Regular prime">regular primes</a>, which feature in <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Kummer's</a> treatment of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Harmonic_analysis">Harmonic analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=19" title="Edit section: Harmonic analysis"><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/Harmonic_analysis" title="Harmonic analysis">Harmonic analysis</a></div> <p>Analysis on Lie groups and certain other groups is called <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a>. <a href="/wiki/Haar_measure" title="Haar measure">Haar measures</a>, that is, integrals invariant under the translation in a Lie group, are used for <a href="/wiki/Pattern_recognition" title="Pattern recognition">pattern recognition</a> and other <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a> techniques.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Combinatorics">Combinatorics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=20" title="Edit section: Combinatorics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a>, the notion of <a href="/wiki/Permutation" title="Permutation">permutation</a> group and the concept of group action are often used to simplify the counting of a set of objects; see in particular <a href="/wiki/Burnside%27s_lemma" title="Burnside's lemma">Burnside's lemma</a>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fifths.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/c/ce/Fifths.png" decoding="async" width="150" height="163" class="mw-file-element" data-file-width="150" data-file-height="163" /></a><figcaption>The circle of fifths may be endowed with a cyclic group structure.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Music">Music</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=21" title="Edit section: Music"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The presence of the 12-<a href="/wiki/Periodic_group" class="mw-redirect" title="Periodic group">periodicity</a> in the <a href="/wiki/Circle_of_fifths" title="Circle of fifths">circle of fifths</a> yields applications of <a href="/wiki/Elementary_group_theory" class="mw-redirect" title="Elementary group theory">elementary group theory</a> in <a href="/wiki/Set_theory_(music)" title="Set theory (music)">musical set theory</a>. <a href="/wiki/Transformational_theory" title="Transformational theory">Transformational theory</a> models musical transformations as elements of a mathematical group. </p> <div class="mw-heading mw-heading3"><h3 id="Physics">Physics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=22" title="Edit section: Physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, groups are important because they describe the symmetries which the laws of physics seem to obey. According to <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>, every continuous symmetry of a physical system corresponds to a <a href="/wiki/Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conservation law</a> of the system. Physicists are very interested in group representations, especially of Lie groups, since these representations often point the way to the "possible" physical theories. Examples of the use of groups in physics include the <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a>, <a href="/wiki/Gauge_theory" title="Gauge theory">gauge theory</a>, the <a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz group</a>, and the <a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré group</a>. </p><p>Group theory can be used to resolve the incompleteness of the statistical interpretations of mechanics developed by <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Willard Gibbs</a>, relating to the summing of an infinite number of probabilities to yield a meaningful solution.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Chemistry_and_materials_science">Chemistry and materials science</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=23" title="Edit section: Chemistry and materials science"><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/Molecular_symmetry" title="Molecular symmetry">Molecular symmetry</a></div> <p>In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> and <a href="/wiki/Materials_science" title="Materials science">materials science</a>, <a href="/wiki/Point_group" title="Point group">point groups</a> are used to classify regular polyhedra, and the <a href="/wiki/Molecular_symmetry" title="Molecular symmetry">symmetries of molecules</a>, and <a href="/wiki/Space_group" title="Space group">space groups</a> to classify <a href="/wiki/Crystal_structure" title="Crystal structure">crystal structures</a>. The assigned groups can then be used to determine physical properties (such as <a href="/wiki/Chemical_polarity" title="Chemical polarity">chemical polarity</a> and <a href="/wiki/Chirality_(chemistry)" title="Chirality (chemistry)">chirality</a>), spectroscopic properties (particularly useful for <a href="/wiki/Raman_spectroscopy" title="Raman spectroscopy">Raman spectroscopy</a>, <a href="/wiki/Infrared_spectroscopy" title="Infrared spectroscopy">infrared spectroscopy</a>, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy, UV/Vis spectroscopy, and fluorescence spectroscopy), and to construct <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular orbitals</a>. </p><p><a href="/wiki/Molecular_symmetry" title="Molecular symmetry">Molecular symmetry</a> is responsible for many physical and spectroscopic properties of compounds and provides relevant information about how chemical reactions occur. In order to assign a point group for any given molecule, it is necessary to find the set of symmetry operations present on it. The symmetry operation is an action, such as a rotation around an axis or a reflection through a mirror plane. In other words, it is an operation that moves the molecule such that it is indistinguishable from the original configuration. In group theory, the rotation axes and mirror planes are called "symmetry elements". These elements can be a point, line or plane with respect to which the symmetry operation is carried out. The symmetry operations of a molecule determine the specific point group for this molecule. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Miri2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Miri2.jpg/100px-Miri2.jpg" decoding="async" width="100" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/d/db/Miri2.jpg 1.5x" data-file-width="124" data-file-height="121" /></a><figcaption>Water molecule with symmetry axis</figcaption></figure> <p>In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, there are five important symmetry operations. They are identity operation (<b>E)</b>, rotation operation or proper rotation (<b>C<sub><i>n</i></sub></b>), reflection operation (<b>σ</b>), inversion (<b>i</b>) and rotation reflection operation or improper rotation (<b>S<sub><i>n</i></sub></b>). The identity operation (<b>E</b>) consists of leaving the molecule as it is. This is equivalent to any number of full rotations around any axis. This is a symmetry of all molecules, whereas the symmetry group of a <a href="/wiki/Chiral" class="mw-redirect" title="Chiral">chiral</a> molecule consists of only the identity operation. An identity operation is a characteristic of every molecule even if it has no symmetry. Rotation around an axis (<b>C<sub><i>n</i></sub></b>) consists of rotating the molecule around a specific axis by a specific angle. It is rotation through the angle 360°/<i>n</i>, where <i>n</i> is an integer, about a rotation axis. For example, if a <a href="/wiki/Water" title="Water">water</a> molecule rotates 180° around the axis that passes through the <a href="/wiki/Oxygen" title="Oxygen">oxygen</a> atom and between the <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a> atoms, it is in the same configuration as it started. In this case, <span class="nowrap"><i>n</i> = 2</span>, since applying it twice produces the identity operation. In molecules with more than one rotation axis, the C<sub>n</sub> axis having the largest value of n is the highest order rotation axis or principal axis. For example in <a href="/wiki/Boron_trifluoride" title="Boron trifluoride">boron trifluoride</a> (BF<sub>3</sub>), the highest order of rotation axis is <b>C<sub>3</sub></b>, so the principal axis of rotation is <b>C<sub>3</sub></b>. </p><p>In the reflection operation (<b>σ</b>) many molecules have mirror planes, although they may not be obvious. The reflection operation exchanges left and right, as if each point had moved perpendicularly through the plane to a position exactly as far from the plane as when it started. When the plane is perpendicular to the principal axis of rotation, it is called <b>σ<sub><i>h</i></sub></b> (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (<b>σ<sub><i>v</i></sub></b>) or dihedral (<b>σ<sub><i>d</i></sub></b>). </p><p>Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from the central point as where it started. Many molecules that seem at first glance to have an inversion center do not; for example, <a href="/wiki/Methane" title="Methane">methane</a> and other <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedral</a> molecules lack inversion symmetry. To see this, hold a methane model with two hydrogen atoms in the vertical plane on the right and two hydrogen atoms in the horizontal plane on the left. Inversion results in two hydrogen atoms in the horizontal plane on the right and two hydrogen atoms in the vertical plane on the left. Inversion is therefore not a symmetry operation of methane, because the orientation of the molecule following the inversion operation differs from the original orientation. And the last operation is improper rotation or rotation reflection operation (<b>S<sub><i>n</i></sub></b>) requires rotation of  360°/<i>n</i>, followed by reflection through a plane perpendicular to the axis of rotation. </p> <div class="mw-heading mw-heading3"><h3 id="Cryptography">Cryptography</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&action=edit&section=24" title="Edit section: Cryptography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Caesar3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Caesar3.svg/220px-Caesar3.svg.png" decoding="async" width="220" height="93" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Caesar3.svg/330px-Caesar3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Caesar3.svg/440px-Caesar3.svg.png 2x" data-file-width="856" data-file-height="361" /></a><figcaption>The <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic group</a> <b>Z</b><sub>26</sub> underlies <a href="/wiki/Caesar%27s_cipher" class="mw-redirect" title="Caesar's cipher">Caesar's cipher</a>.</figcaption></figure> <p>Very large groups of prime order constructed in <a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">elliptic curve cryptography</a> serve for <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>. Cryptographical methods of this kind benefit from the flexibility of the geometric objects, hence their group structures, together with the complicated structure of these groups, which make the <a href="/wiki/Discrete_logarithm" title="Discrete logarithm">discrete logarithm</a> very hard to calculate. One of the earliest encryption protocols, <a href="/wiki/Caesar_cipher" title="Caesar cipher">Caesar's cipher</a>, may also be interpreted as a (very easy) group operation. Most cryptographic schemes use groups in some way. In particular <a href="/wiki/Diffie%E2%80%93Hellman_key_exchange" title="Diffie–Hellman key exchange">Diffie–Hellman key exchange</a> uses finite <a href="/wiki/Cyclic_group" title="Cyclic group">cyclic groups</a>. So the term <a href="/wiki/Group-based_cryptography" title="Group-based cryptography">group-based cryptography</a> refers mostly to <a href="/wiki/Cryptographic_protocol" title="Cryptographic protocol">cryptographic protocols</a> that use infinite <a href="/wiki/Non-abelian_group" title="Non-abelian group">non-abelian groups</a> such as a <a href="/wiki/Braid_group" title="Braid group">braid group</a>. </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=Group_theory&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li> <li><a href="/wiki/Examples_of_groups" title="Examples of groups">Examples of groups</a></li> <li><a href="/wiki/Bass-Serre_theory" class="mw-redirect" title="Bass-Serre theory">Bass-Serre theory</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=Group_theory&action=edit&section=26" 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"><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="CITEREFElwes2006" class="citation cs2">Elwes, Richard (December 2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090202092008/http://plus.maths.org/issue41/features/elwes/index.html">"An enormous theorem: the classification of finite simple groups"</a>, <i><a href="/wiki/Plus_Magazine" title="Plus Magazine">Plus Magazine</a></i> (41), archived from <a rel="nofollow" class="external text" href="http://plus.maths.org/issue41/features/elwes/index.html">the original</a> on 2009-02-02<span class="reference-accessdate">, retrieved <span class="nowrap">2011-12-20</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Plus+Magazine&rft.atitle=An+enormous+theorem%3A+the+classification+of+finite+simple+groups&rft.issue=41&rft.date=2006-12&rft.aulast=Elwes&rft.aufirst=Richard&rft_id=http%3A%2F%2Fplus.maths.org%2Fissue41%2Ffeatures%2Felwes%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">This process of imposing extra structure has been formalized through the notion of a <a href="/wiki/Group_object" title="Group object">group object</a> in a suitable <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>. Thus Lie groups are group objects in the category of differentiable manifolds and affine algebraic groups are group objects in the category of affine algebraic varieties.</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">Such as <a href="/wiki/Group_cohomology" title="Group cohomology">group cohomology</a> or <a href="/wiki/Equivariant_algebraic_K-theory" title="Equivariant algebraic K-theory">equivariant K-theory</a>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">In particular, if the representation is <a href="/wiki/Faithful_representation" title="Faithful representation">faithful</a>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArthur_Tresse1893" class="citation cs2">Arthur Tresse (1893), <a rel="nofollow" class="external text" href="https://zenodo.org/record/2273334">"Sur les invariants différentiels des groupes continus de transformations"</a>, <i>Acta Mathematica</i>, <b>18</b>: 1–88, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02418270">10.1007/bf02418270</a></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=Sur+les+invariants+diff%C3%A9rentiels+des+groupes+continus+de+transformations&rft.volume=18&rft.pages=1-88&rft.date=1893&rft_id=info%3Adoi%2F10.1007%2Fbf02418270&rft.au=Arthur+Tresse&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F2273334&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFSchuppLyndon2001">Schupp & Lyndon 2001</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFLa_Harpe2000">La Harpe 2000</a></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">See the <a href="/wiki/Birch_and_Swinnerton-Dyer_conjecture" title="Birch and Swinnerton-Dyer conjecture">Birch and Swinnerton-Dyer conjecture</a>, one of the <a href="/wiki/Millennium_problem" class="mw-redirect" title="Millennium problem">millennium problems</a></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="CITEREFAbramovichKaruMatsukiWlodarczyk2002" class="citation cs2">Abramovich, Dan; Karu, Kalle; Matsuki, Kenji; Wlodarczyk, Jaroslaw (2002), "Torification and factorization of birational maps", <i><a href="/wiki/Journal_of_the_American_Mathematical_Society" title="Journal of the American Mathematical Society">Journal of the American Mathematical Society</a></i>, <b>15</b> (3): 531–572, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9904135">math/9904135</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0894-0347-02-00396-X">10.1090/S0894-0347-02-00396-X</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1896232">1896232</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18211120">18211120</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+American+Mathematical+Society&rft.atitle=Torification+and+factorization+of+birational+maps&rft.volume=15&rft.issue=3&rft.pages=531-572&rft.date=2002&rft_id=info%3Aarxiv%2Fmath%2F9904135&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1896232%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18211120%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1090%2FS0894-0347-02-00396-X&rft.aulast=Abramovich&rft.aufirst=Dan&rft.au=Karu%2C+Kalle&rft.au=Matsuki%2C+Kenji&rft.au=Wlodarczyk%2C+Jaroslaw&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" 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="CITEREFLenz1990" class="citation cs2">Lenz, Reiner (1990), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/grouptheoretical0000lenz"><i>Group theoretical methods in image processing</i></a></span>, Lecture Notes in Computer Science, vol. 413, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-52290-5">10.1007/3-540-52290-5</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-52290-6" title="Special:BookSources/978-0-387-52290-6"><bdi>978-0-387-52290-6</bdi></a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2738874">2738874</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+theoretical+methods+in+image+processing&rft.place=Berlin%2C+New+York&rft.series=Lecture+Notes+in+Computer+Science&rft.pub=Springer-Verlag&rft.date=1990&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A2738874%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2F3-540-52290-5&rft.isbn=978-0-387-52290-6&rft.aulast=Lenz&rft.aufirst=Reiner&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgrouptheoretical0000lenz&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" 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="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a>, Cybernetics: Or Control and Communication in the Animal and the Machine, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0262730099" title="Special:BookSources/978-0262730099">978-0262730099</a>, Ch 2</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=Group_theory&action=edit&section=27" 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="CITEREFBorel1991" class="citation cs2"><a href="/wiki/Armand_Borel" title="Armand Borel">Borel, Armand</a> (1991), <i>Linear algebraic groups</i>, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0941-6">10.1007/978-1-4612-0941-6</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97370-8" title="Special:BookSources/978-0-387-97370-8"><bdi>978-0-387-97370-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1102012">1102012</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+algebraic+groups&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1991&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1102012%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0941-6&rft.isbn=978-0-387-97370-8&rft.aulast=Borel&rft.aufirst=Armand&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarter2009" class="citation cs2">Carter, Nathan C. (2009), <a rel="nofollow" class="external text" href="http://web.bentley.edu/empl/c/ncarter/vgt/"><i>Visual group theory</i></a>, Classroom Resource Materials Series, <a href="/wiki/Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-757-1" title="Special:BookSources/978-0-88385-757-1"><bdi>978-0-88385-757-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2504193">2504193</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Visual+group+theory&rft.series=Classroom+Resource+Materials+Series&rft.pub=Mathematical+Association+of+America&rft.date=2009&rft.isbn=978-0-88385-757-1&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2504193%23id-name%3DMR&rft.aulast=Carter&rft.aufirst=Nathan+C.&rft_id=http%3A%2F%2Fweb.bentley.edu%2Fempl%2Fc%2Fncarter%2Fvgt%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCannon1969" class="citation cs2">Cannon, John J. (1969), "Computers in group theory: A survey", <i>Communications of the ACM</i>, <b>12</b>: 3–12, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F362835.362837">10.1145/362835.362837</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0290613">0290613</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18226463">18226463</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Communications+of+the+ACM&rft.atitle=Computers+in+group+theory%3A+A+survey&rft.volume=12&rft.pages=3-12&rft.date=1969&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0290613%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18226463%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F362835.362837&rft.aulast=Cannon&rft.aufirst=John+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrucht1939" class="citation cs2">Frucht, R. (1939), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20081201083831/http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0">"Herstellung von Graphen mit vorgegebener abstrakter Gruppe"</a>, <i>Compositio Mathematica</i>, <b>6</b>: 239–50, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0010-437X">0010-437X</a>, archived from <a rel="nofollow" class="external text" href="http://www.numdam.org/numdam-bin/fitem?id=CM_1939__6__239_0">the original</a> on 2008-12-01</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Compositio+Mathematica&rft.atitle=Herstellung+von+Graphen+mit+vorgegebener+abstrakter+Gruppe&rft.volume=6&rft.pages=239-50&rft.date=1939&rft.issn=0010-437X&rft.aulast=Frucht&rft.aufirst=R.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fnumdam-bin%2Ffitem%3Fid%3DCM_1939__6__239_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubitskyStewart2006" class="citation cs2"><a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Golubitsky, Martin</a>; Stewart, Ian (2006), "Nonlinear dynamics of networks: the groupoid formalism", <i>Bull. Amer. Math. Soc. (N.S.)</i>, <b>43</b> (3): 305–364, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0273-0979-06-01108-6">10.1090/S0273-0979-06-01108-6</a></span>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2223010">2223010</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.+%28N.S.%29&rft.atitle=Nonlinear+dynamics+of+networks%3A+the+groupoid+formalism&rft.volume=43&rft.issue=3&rft.pages=305-364&rft.date=2006&rft_id=info%3Adoi%2F10.1090%2FS0273-0979-06-01108-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2223010%23id-name%3DMR&rft.aulast=Golubitsky&rft.aufirst=Martin&rft.au=Stewart%2C+Ian&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span> Shows the advantage of generalising from group to <a href="/wiki/Groupoid" title="Groupoid">groupoid</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJudson1997" class="citation cs2">Judson, Thomas W. (1997), <a rel="nofollow" class="external text" href="http://abstract.ups.edu"><i>Abstract Algebra: Theory and Applications</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra%3A+Theory+and+Applications&rft.date=1997&rft.aulast=Judson&rft.aufirst=Thomas+W.&rft_id=http%3A%2F%2Fabstract.ups.edu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span> An introductory undergraduate text in the spirit of texts by Gallian or Herstein, covering groups, rings, integral domains, fields and Galois theory. Free downloadable PDF with open-source <a href="/wiki/GNU_Free_Documentation_License" title="GNU Free Documentation License">GFDL</a> license.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1986" class="citation cs2">Kleiner, Israel (1986), "The evolution of group theory: a brief survey", <i><a href="/wiki/Mathematics_Magazine" title="Mathematics Magazine">Mathematics Magazine</a></i>, <b>59</b> (4): 195–215, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2690312">10.2307/2690312</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-570X">0025-570X</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2690312">2690312</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0863090">0863090</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=The+evolution+of+group+theory%3A+a+brief+survey&rft.volume=59&rft.issue=4&rft.pages=195-215&rft.date=1986&rft.issn=0025-570X&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D863090%23id-name%3DMR&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2690312%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2690312&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLa_Harpe2000" class="citation cs2">La Harpe, Pierre de (2000), <i>Topics in geometric group theory</i>, <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-31721-2" title="Special:BookSources/978-0-226-31721-2"><bdi>978-0-226-31721-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+geometric+group+theory&rft.pub=University+of+Chicago+Press&rft.date=2000&rft.isbn=978-0-226-31721-2&rft.aulast=La+Harpe&rft.aufirst=Pierre+de&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLivio,_M.2005" class="citation cs2"><a href="/wiki/Mario_Livio" title="Mario Livio">Livio, M.</a> (2005), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/equationthatcoul0000livi"><i>The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry</i></a></span>, Simon & Schuster, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7432-5820-7" title="Special:BookSources/0-7432-5820-7"><bdi>0-7432-5820-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Equation+That+Couldn%27t+Be+Solved%3A+How+Mathematical+Genius+Discovered+the+Language+of+Symmetry&rft.pub=Simon+%26+Schuster&rft.date=2005&rft.isbn=0-7432-5820-7&rft.au=Livio%2C+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fequationthatcoul0000livi&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span> Conveys the practical value of group theory by explaining how it points to <a href="/wiki/Symmetries" class="mw-redirect" title="Symmetries">symmetries</a> in <a href="/wiki/Physics" title="Physics">physics</a> and other sciences.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMumford1970" class="citation cs2"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a> (1970), <i>Abelian varieties</i>, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-560528-0" title="Special:BookSources/978-0-19-560528-0"><bdi>978-0-19-560528-0</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/138290">138290</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abelian+varieties&rft.pub=Oxford+University+Press&rft.date=1970&rft_id=info%3Aoclcnum%2F138290&rft.isbn=978-0-19-560528-0&rft.aulast=Mumford&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><a href="/wiki/Mark_Ronan" title="Mark Ronan">Ronan M.</a>, 2006. <i>Symmetry and the Monster</i>. Oxford University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-280722-6" title="Special:BookSources/0-19-280722-6">0-19-280722-6</a>. For lay readers. Describes the quest to find the basic building blocks for finite groups.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman,_Joseph1994" class="citation cs2">Rotman, Joseph (1994), <i>An introduction to the theory of groups</i>, New York: Springer-Verlag, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94285-8" title="Special:BookSources/0-387-94285-8"><bdi>0-387-94285-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+theory+of+groups&rft.place=New+York&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=0-387-94285-8&rft.au=Rotman%2C+Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span> A standard contemporary reference.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchuppLyndon2001" class="citation cs2"><a href="/wiki/Paul_Schupp" title="Paul Schupp">Schupp, Paul E.</a>; <a href="/wiki/Roger_Lyndon" title="Roger Lyndon">Lyndon, Roger C.</a> (2001), <i>Combinatorial group theory</i>, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-41158-1" title="Special:BookSources/978-3-540-41158-1"><bdi>978-3-540-41158-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+group+theory&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.date=2001&rft.isbn=978-3-540-41158-1&rft.aulast=Schupp&rft.aufirst=Paul+E.&rft.au=Lyndon%2C+Roger+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScott,_W._R.1987" class="citation cs2">Scott, W. R. (1987) [1964], <i>Group Theory</i>, New York: Dover, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-65377-3" title="Special:BookSources/0-486-65377-3"><bdi>0-486-65377-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Group+Theory&rft.place=New+York&rft.pub=Dover&rft.date=1987&rft.isbn=0-486-65377-3&rft.au=Scott%2C+W.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span> Inexpensive and fairly readable, but somewhat dated in emphasis, style, and notation.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShatz1972" class="citation cs2">Shatz, Stephen S. (1972), <i>Profinite groups, arithmetic, and geometry</i>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-08017-8" title="Special:BookSources/978-0-691-08017-8"><bdi>978-0-691-08017-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0347778">0347778</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Profinite+groups%2C+arithmetic%2C+and+geometry&rft.pub=Princeton+University+Press&rft.date=1972&rft.isbn=978-0-691-08017-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0347778%23id-name%3DMR&rft.aulast=Shatz&rft.aufirst=Stephen+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeibel1994" class="citation book cs2"><a href="/wiki/Charles_Weibel" title="Charles Weibel">Weibel, Charles A.</a> (1994), <i>An introduction to homological algebra</i>, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55987-4" title="Special:BookSources/978-0-521-55987-4"><bdi>978-0-521-55987-4</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1269324">1269324</a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36131259">36131259</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+homological+algebra&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=info%3Aoclcnum%2F36131259&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1269324%23id-name%3DMR&rft.isbn=978-0-521-55987-4&rft.aulast=Weibel&rft.aufirst=Charles+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span></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=Group_theory&action=edit&section=28" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output 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class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/20px-Commons-logo.svg.png" decoding="async" width="20" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/40px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></span><span class="sister-link"><a href="https://commons.wikimedia.org/wiki/Category:Group_theory" class="extiw" title="c:Category:Group theory">Media</a> from Commons</span></li><li><span class="sister-logo"><span class="mw-valign-middle" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/23px-Wikiquote-logo.svg.png" decoding="async" width="23" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/35px-Wikiquote-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikiquote-logo.svg/46px-Wikiquote-logo.svg.png 2x" data-file-width="300" data-file-height="355" /></span></span></span><span class="sister-link"><a href="https://en.wikiquote.org/wiki/Group_theory" class="extiw" title="q:Group theory">Quotations</a> from Wikiquote</span></li></ul></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Abstract_groups.html">History of the abstract group concept</a></li> <li><a rel="nofollow" class="external text" href="https://archive.today/20120723235509/http://www.bangor.ac.uk/r.brown/hdaweb2.htm">Higher dimensional group theory</a> This presents a view of group theory as level one of a theory that extends in all dimensions, and has applications in homotopy theory and to higher dimensional nonabelian methods for local-to-global problems.</li> <li><a rel="nofollow" class="external text" href="http://plus.maths.org/issue48/package/index.html">Plus teacher and student package: Group Theory</a> This package brings together all the articles on group theory from <i>Plus</i>, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge, exploring applications and recent breakthroughs, and giving explicit definitions and examples of groups.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurnside1911" class="citation encyclopaedia cs2"><a href="/wiki/William_Burnside" title="William Burnside">Burnside, William</a> (1911), <span class="cs1-ws-icon" title="s:1911 Encyclopædia Britannica/Groups, Theory of"><a class="external text" href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Groups,_Theory_of">"Groups, Theory of" </a></span>, in <a href="/wiki/Hugh_Chisholm" title="Hugh Chisholm">Chisholm, Hugh</a> (ed.), <i><a href="/wiki/Encyclop%C3%A6dia_Britannica_Eleventh_Edition" title="Encyclopædia Britannica Eleventh Edition">Encyclopædia Britannica</a></i>, vol. 12 (11th ed.), Cambridge University Press, pp. 626–636</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Groups%2C+Theory+of&rft.btitle=Encyclop%C3%A6dia+Britannica&rft.pages=626-636&rft.edition=11th&rft.pub=Cambridge+University+Press&rft.date=1911&rft.aulast=Burnside&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+theory" class="Z3988"></span> This is a detailed exposition of contemporaneous understanding of Group Theory by an early researcher in the field.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Major_mathematics_areas" style="font-size:114%;margin:0 4em">Major <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> areas</div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_mathematics" title="History of mathematics">History</a> <ul><li><a href="/wiki/Timeline_of_mathematics" title="Timeline of mathematics">Timeline</a></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future</a></li></ul></li> <li><a href="/wiki/Lists_of_mathematics_topics" title="Lists of mathematics topics">Lists</a></li> <li><a href="/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of 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