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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" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Group_theory" title="Group theory – English" lang="en" hreflang="en" data-title="Group theory" data-language-autonym="English" data-language-local-name="English" class="interlanguage-link-target"><span>English</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 – 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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" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> and <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, <b>group theory</b> studies a type of <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> called a <b><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a></b>. Group theory is often used in mathematics as a starting point for the study of many algebraic structures, such as a set of numbers along with its <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>. Because group theory is also useful for studying <a href="/wiki/Symmetry" title="Symmetry">symmetry</a> in <a href="/wiki/Nature" title="Nature">nature</a> and abstract systems, it has many applications in <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&veaction=edit&section=1" title="Change section: Definition" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Group_theory&action=edit&section=1" title="Edit section's source code: Definition"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A group is a <a href="/wiki/Set" title="Set">set</a> (collection) <i>G</i> whose members are called <i>elements</i>. The elements can be numbers of some kind, or other abstract objects. The elements can even be material objects. There is also a <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> which combines any two elements of <i>G</i> and results in another element of <i>G</i>. This new element may be different from either of the original two, but it does not have to be: it only has to be an element of <i>G</i>. To be a group, the following four conditions must be true of <i>G</i> and the operation defined over <i>G</i>: </p> <ul><li><i>Closure</i>: It is required to check if the suggested operation is actually an operation on the set. If an operation is used on any elements in the group, the element that is formed will also be part of the group. <ul><li>For all <i>a</i>, <i>b</i> in <i>G</i>, the result of the operation <i>a</i> • <i>b</i> is also in <i>G</i>.</li></ul></li> <li><i>Identity element</i>: One element of the group is special. It is called the identity element, often written as <i>e</i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> If the operation is used with the identity element and another element, the other element will not change. <ul><li>There exists an element <i>e</i> in <i>G</i>, such that for all elements <i>a</i> in <i>G</i>, the equation </li></ul></li></ul> <p><span style="white-space:nowrap;"><i>e</i> • <i>a</i> = <i>a</i> • <i>e</i> = <i>a</i></span> holds. </p> <ul><li><i>Associativity</i>: When there are many operations, it does not matter in what order they are done, the result will be the same. <ul><li>For all <i>a</i>, <i>b</i> and <i>c</i> in <i>G</i>, the equation (<i>a</i> • <i>b</i>) • <i>c</i> = <i>a</i> • (<i>b</i> • <i>c</i>) holds.</li></ul></li> <li><i>Inverse element</i>: Every element in the group has another element in the group known as its inverse. When the operation is performed between them, the result is the identity element. <ul><li>For each <i>a</i> in <i>G</i>, there exists an element <i>b</i> in <i>G</i> such that <i>a</i> • <i>b</i> = <i>b</i> • <i>a</i> = <i>e</i>, where <i>e</i> is the identity element.</li></ul></li></ul> <p>An important property about groups is whether the order of the elements matters when two elements are combined using the operation. Switching the order of two combined elements typically gives a different result. This means that we have many cases with <i>a</i> • <i>b</i> ≠ <i>b</i> • <i>a</i>. However, if the order of a and b actually does not matter (for <i>any</i> a and b, we <i>always</i> have <i>a</i> • <i>b</i> = <i>b</i> • <i>a</i>), then the group is called an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> (or commutative group).<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>If a group contains finitely many elements, then it is called a finite group. If a <a href="/wiki/Subset" title="Subset">subset</a> of a group is also a group, then the subset is called a subgroup.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&veaction=edit&section=2" title="Change section: Examples" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Group_theory&action=edit&section=2" title="Edit section's source code: Examples"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simple example of a group is the set of <a href="/wiki/Integer" title="Integer">integers</a> (called <b>Z</b>) with the addition operator +. This group is called <i>G</i> = (<b>Z</b>, +). Each of the required conditions is true of <i>G</i>, so <i>G</i> is a group. If we replace • with the + sign, then we can easily check that these conditions are true: </p> <ul><li>Closure: For any two integers <i>a</i> and <i>b</i> in <i>G</i>, <i>a</i> + <i>b</i> is also in <i>G</i>. For example, 3 and 5 are integers; and if we add them together, we see that 8 is an integer too.</li> <li>Identity element: For any integer <i>a</i> in <i>G</i>, <i>a</i> + 0 = <i>a</i>. So 0 is the identity element of G. For example, 69+0 is still 69.</li> <li>Associativity: For any three integers <i>a</i>, <i>b</i>, and <i>c</i> in <i>G</i>, (<i>a</i> + <i>b</i>) + <i>c</i> = <i>a</i> + (<i>b</i> + <i>c</i>). For example, (1+3)+7 is the same as 1+(3+7).</li> <li>Inverse element: For any integer <i>a</i> in <i>G</i>, there is another integer <i>b</i> = −<i>a</i> in <i>G</i> such that <i>a</i> + <i>b</i> = 0. For example, 4 + (−4) = 0. In other words, −4 is the <i><a href="/wiki/Inverse_element" title="Inverse element">inverse element</a></i> of 4.</li></ul> <p>The addition operator is also <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>: the order of <i>a</i> and <i>b</i> does not matter. For any two integers <i>a</i> and <i>b</i> in G, <i>a</i> + <i>b</i> = <i>b</i> + <i>a</i>. For example, 7+3 is the same as 3+7. Thus, <i>G</i> is also an abelian group. </p><p>Commutative operators are common in our daily lives. For example, 7 · 3 is also the same as 3 · 7. However, not all <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> are commutative. For example, 7÷3 is different from 3÷7 (and neither of these results is an integer), and 7<sup>3</sup> is different from 3<sup>7</sup>. Therefore, not all combinations of number sets and mathematical operations can form an abelian group—or even a group at all. </p> <div class="mw-heading mw-heading2"><h2 id="Related_pages">Related pages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_theory&veaction=edit&section=3" title="Change section: Related pages" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Group_theory&action=edit&section=3" title="Edit section's source code: Related pages"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></li></ul> <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&veaction=edit&section=4" title="Change section: References" class="mw-editsection-visualeditor"><span>change</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Group_theory&action=edit&section=4" title="Edit section's source code: References"><span>change source</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r9724332">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathvault.ca/hub/higher-math/math-symbols/algebra-symbols/">"Comprehensive List of Algebra Symbols"</a>. <i>Math Vault</i>. 2020-03-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2020-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+Vault&rft.atitle=Comprehensive+List+of+Algebra+Symbols&rft.date=2020-03-25&rft_id=https%3A%2F%2Fmathvault.ca%2Fhub%2Fhigher-math%2Fmath-symbols%2Falgebra-symbols%2F&rfr_id=info%3Asid%2Fsimple.wikipedia.org%3AGroup+theory" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r9724332"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/group-theory">"group theory | Definition, Axioms, & Applications"</a>. <i>Encyclopedia Britannica</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+Britannica&rft.atitle=group+theory+%7C+Definition%2C+Axioms%2C+%26+Applications&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fgroup-theory&rfr_id=info%3Asid%2Fsimple.wikipedia.org%3AGroup+theory" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r9724332"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Group.html">"Group"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2020-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Group&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FGroup.html&rfr_id=info%3Asid%2Fsimple.wikipedia.org%3AGroup+theory" class="Z3988"></span></span> </li> </ol> <p><br /> </p> <div class="boilerplate metadata plainlinks" id="stub"><span typeof="mw:File"><a href="/wiki/File:Xy_icon.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Xy_icon.svg/30px-Xy_icon.svg.png" decoding="async" width="30" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Xy_icon.svg/45px-Xy_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Xy_icon.svg/60px-Xy_icon.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <i>This <a href="/wiki/Wikipedia:Stub" title="Wikipedia:Stub">short article</a> about <a href="/wiki/Category:Math_stubs" title="Category:Math stubs">mathematics</a> can be made longer. 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