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id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Início</div> </a> </li> <li id="toc-Definição" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definição"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definição</span> </div> </a> <button aria-controls="toc-Definição-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>Alternar a subsecção Definição</span> </button> <ul id="toc-Definição-sublist" class="vector-toc-list"> <li id="toc-Ordem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordem"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Ordem</span> </div> </a> <ul id="toc-Ordem-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Exemplos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Exemplos"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Exemplos</span> </div> </a> <button aria-controls="toc-Exemplos-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>Alternar a subsecção Exemplos</span> </button> <ul id="toc-Exemplos-sublist" class="vector-toc-list"> <li id="toc-Grupo_de_simetrias_de_um_quadrado" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grupo_de_simetrias_de_um_quadrado"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Grupo de simetrias de um quadrado</span> </div> </a> <ul id="toc-Grupo_de_simetrias_de_um_quadrado-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grupo_das_permutações" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grupo_das_permutações"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Grupo das permutações</span> </div> </a> <ul id="toc-Grupo_das_permutações-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Propriedades_Imediatas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Propriedades_Imediatas"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Propriedades Imediatas</span> </div> </a> <ul id="toc-Propriedades_Imediatas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Conceitos_básicos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Conceitos_básicos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Conceitos básicos</span> </div> </a> <button aria-controls="toc-Conceitos_básicos-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>Alternar a subsecção Conceitos básicos</span> </button> <ul id="toc-Conceitos_básicos-sublist" class="vector-toc-list"> <li id="toc-Homomorfismos_de_grupos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homomorfismos_de_grupos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Homomorfismos de grupos</span> </div> </a> <ul id="toc-Homomorfismos_de_grupos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Exemplos_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exemplos_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Exemplos</span> </div> </a> <ul id="toc-Exemplos_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propriedades" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Propriedades"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Propriedades</span> </div> </a> <ul id="toc-Propriedades-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tipos_de_homomorfismos" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tipos_de_homomorfismos"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Tipos de homomorfismos</span> </div> </a> <ul id="toc-Tipos_de_homomorfismos-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Subgrupos" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subgrupos"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Subgrupos</span> </div> </a> <ul id="toc-Subgrupos-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ver_também" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ver_também"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Ver também</span> </div> </a> <ul id="toc-Ver_também-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="Conteúdo" 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="Alternar o índice" > <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">Alternar o índice</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">Grupo (matemática)</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="Ir para um artigo noutra língua. Disponível em 83 línguas" > <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-83" 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">83 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Groep_(wiskunde)" title="Groep (wiskunde) — africanês" lang="af" hreflang="af" data-title="Groep (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="africanês" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="زمرة (رياضيات) — árabe" lang="ar" hreflang="ar" data-title="زمرة (رياضيات)" data-language-autonym="العربية" data-language-local-name="árabe" class="interlanguage-link-target"><span>العربية</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%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Төркөм (математика) — bashkir" lang="ba" hreflang="ba" data-title="Төркөм (математика)" 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%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Група (алгебра) — bielorrusso" lang="be" hreflang="be" data-title="Група (алгебра)" data-language-autonym="Беларуская" data-language-local-name="bielorrusso" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Група (алгебра) — búlgaro" lang="bg" hreflang="bg" data-title="Група (алгебра)" data-language-autonym="Български" data-language-local-name="búlgaro" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A7%81%E0%A6%AA_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="গ্রুপ (গণিত) — bengalês" lang="bn" hreflang="bn" data-title="গ্রুপ (গণিত)" data-language-autonym="বাংলা" data-language-local-name="bengalês" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_(matem%C3%A0tiques)" title="Grup (matemàtiques) — catalão" lang="ca" hreflang="ca" data-title="Grup (matemàtiques)" data-language-autonym="Català" data-language-local-name="catalão" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%88%D9%BE_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="گرووپ (ماتماتیک) — curdo central" lang="ckb" hreflang="ckb" data-title="گرووپ (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="curdo central" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="artigo bom"><a href="https://cs.wikipedia.org/wiki/Grupa" title="Grupa — checo" lang="cs" hreflang="cs" data-title="Grupa" data-language-autonym="Čeština" data-language-local-name="checo" class="interlanguage-link-target"><span>Čeština</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_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ушкăн (математика) — chuvash" lang="cv" hreflang="cv" data-title="Ушкăн (математика)" data-language-autonym="Чӑвашла" data-language-local-name="chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gr%C5%B5p_(mathemateg)" title="Grŵp (mathemateg) — galês" lang="cy" hreflang="cy" data-title="Grŵp (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="galês" 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/Gruppe_(matematik)" title="Gruppe (matematik) — dinamarquês" lang="da" hreflang="da" data-title="Gruppe (matematik)" data-language-autonym="Dansk" data-language-local-name="dinamarquês" 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/Gruppe_(Mathematik)" title="Gruppe (Mathematik) — alemão" lang="de" hreflang="de" data-title="Gruppe (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="alemão" 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%9F%CE%BC%CE%AC%CE%B4%CE%B1" title="Ομάδα — grego" lang="el" hreflang="el" data-title="Ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="grego" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437796 badge-featuredarticle mw-list-item" title="artigo destacado"><a href="https://en.wikipedia.org/wiki/Group_(mathematics)" title="Group (mathematics) — inglês" lang="en" hreflang="en" data-title="Group (mathematics)" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grupo_(algebro)" title="Grupo (algebro) — esperanto" lang="eo" hreflang="eo" data-title="Grupo (algebro)" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_(matem%C3%A1tica)" title="Grupo (matemática) — espanhol" lang="es" hreflang="es" data-title="Grupo (matemática)" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/R%C3%BChm_(matemaatika)" title="Rühm (matemaatika) — estónio" lang="et" hreflang="et" data-title="Rühm (matemaatika)" data-language-autonym="Eesti" data-language-local-name="estónio" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde_(matematika)" title="Talde (matematika) — basco" lang="eu" hreflang="eu" data-title="Talde (matematika)" data-language-autonym="Euskara" data-language-local-name="basco" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="گروه (ریاضیات) — persa" lang="fa" hreflang="fa" data-title="گروه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="persa" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ryhm%C3%A4_(algebra)" title="Ryhmä (algebra) — finlandês" lang="fi" hreflang="fi" data-title="Ryhmä (algebra)" data-language-autonym="Suomi" data-language-local-name="finlandês" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques) — francês" lang="fr" hreflang="fr" data-title="Groupe (mathématiques)" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Sk%C3%B6%C3%B6l_(Matematiik)" title="Skööl (Matematiik) — frísio setentrional" lang="frr" hreflang="frr" data-title="Skööl (Matematiik)" data-language-autonym="Nordfriisk" data-language-local-name="frísio setentrional" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Gr%C3%BApa_(matamaitic)" title="Grúpa (matamaitic) — irlandês" lang="ga" hreflang="ga" data-title="Grúpa (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="irlandês" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Grupo_(matem%C3%A1ticas)" title="Grupo (matemáticas) — galego" lang="gl" hreflang="gl" data-title="Grupo (matemáticas)" data-language-autonym="Galego" data-language-local-name="galego" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="חבורה (מבנה אלגברי) — hebraico" lang="he" hreflang="he" data-title="חבורה (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</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%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="समूह (गणितशास्त्र) — hindi" lang="hi" hreflang="hi" data-title="समूह (गणितशास्त्र)" 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/Grupa_(matematika)" title="Grupa (matematika) — croata" lang="hr" hreflang="hr" data-title="Grupa (matematika)" data-language-autonym="Hrvatski" data-language-local-name="croata" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Csoport_(matematika)" title="Csoport (matematika) — húngaro" lang="hu" hreflang="hu" data-title="Csoport (matematika)" data-language-autonym="Magyar" data-language-local-name="húngaro" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BD%D5%B8%D6%82%D5%B4%D5%A2_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Խումբ (մաթեմատիկա) — arménio" lang="hy" hreflang="hy" data-title="Խումբ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="arménio" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_(mathematica)" title="Gruppo (mathematica) — interlíngua" lang="ia" hreflang="ia" data-title="Gruppo (mathematica)" data-language-autonym="Interlingua" data-language-local-name="interlíngua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_(matematika)" title="Grup (matematika) — indonésio" lang="id" hreflang="id" data-title="Grup (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonésio" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Gr%C3%BApa" title="Grúpa — islandês" lang="is" hreflang="is" data-title="Grúpa" data-language-autonym="Íslenska" data-language-local-name="islandês" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_(matematica)" title="Gruppo (matematica) — italiano" lang="it" hreflang="it" data-title="Gruppo (matematica)" data-language-autonym="Italiano" data-language-local-name="italiano" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%BE%A4_(%E6%95%B0%E5%AD%A6)" title="群 (数学) — japonês" lang="ja" hreflang="ja" data-title="群 (数学)" data-language-autonym="日本語" data-language-local-name="japonês" class="interlanguage-link-target"><span>日本語</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%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ჯგუფი (მათემატიკა) — georgiano" lang="ka" hreflang="ka" data-title="ჯგუფი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="georgiano" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tagrumma_(tusnakt)" title="Tagrumma (tusnakt) — kabyle" lang="kab" hreflang="kab" data-title="Tagrumma (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="kabyle" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Топ (математика) — cazaque" lang="kk" hreflang="kk" data-title="Топ (математика)" data-language-autonym="Қазақша" data-language-local-name="cazaque" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%8D%E0%B2%B0%E0%B3%82%E0%B2%AA%E0%B3%8D" title="ಗ್ರೂಪ್ — canarim" lang="kn" hreflang="kn" data-title="ಗ್ರೂಪ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="canarim" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0_(%EC%88%98%ED%95%99)" title="군 (수학) — coreano" lang="ko" hreflang="ko" data-title="군 (수학)" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Caterva_(mathematica)" title="Caterva (mathematica) — latim" lang="la" hreflang="la" data-title="Caterva (mathematica)" data-language-autonym="Latina" data-language-local-name="latim" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Grupp_(Algeber)" title="Grupp (Algeber) — luxemburguês" lang="lb" hreflang="lb" data-title="Grupp (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="luxemburguês" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Grupp_(matem%C3%A0tica)" title="Grupp (matemàtica) — lombardo" lang="lmo" hreflang="lmo" data-title="Grupp (matemàtica)" data-language-autonym="Lombard" data-language-local-name="lombardo" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Grup%C4%97_(algebra)" title="Grupė (algebra) — lituano" lang="lt" hreflang="lt" data-title="Grupė (algebra)" data-language-autonym="Lietuvių" data-language-local-name="lituano" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Grupa_(matem%C4%81tika)" title="Grupa (matemātika) — letão" lang="lv" hreflang="lv" data-title="Grupa (matemātika)" data-language-autonym="Latviešu" data-language-local-name="letão" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Vory_(matematika)" title="Vory (matematika) — malgaxe" lang="mg" hreflang="mg" data-title="Vory (matematika)" data-language-autonym="Malagasy" data-language-local-name="malgaxe" class="interlanguage-link-target"><span>Malagasy</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" title="ഗ്രൂപ്പ് — malaiala" lang="ml" hreflang="ml" data-title="ഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="malaiala" 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/Kumpulan_(matematik)" title="Kumpulan (matematik) — malaio" lang="ms" hreflang="ms" data-title="Kumpulan (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="malaio" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Grupp_(matematika)" title="Grupp (matematika) — maltês" lang="mt" hreflang="mt" data-title="Grupp (matematika)" data-language-autonym="Malti" data-language-local-name="maltês" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Groep_(wiskunde)" title="Groep (wiskunde) — neerlandês" lang="nl" hreflang="nl" data-title="Groep (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="neerlandês" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_gruppe" title="Matematisk gruppe — norueguês nynorsk" lang="nn" hreflang="nn" data-title="Matematisk gruppe" data-language-autonym="Norsk nynorsk" data-language-local-name="norueguês nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gruppe_(matematikk)" title="Gruppe (matematikk) — norueguês bokmål" lang="nb" hreflang="nb" data-title="Gruppe (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norueguês bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Grupe_(matematike)" title="Grupe (matematike) — Novial" lang="nov" hreflang="nov" data-title="Grupe (matematike)" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Grop_(matematicas)" title="Grop (matematicas) — occitano" lang="oc" hreflang="oc" data-title="Grop (matematicas)" data-language-autonym="Occitan" data-language-local-name="occitano" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_(matematyka)" title="Grupa (matematyka) — polaco" lang="pl" hreflang="pl" data-title="Grupa (matematyka)" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Strop" title="Strop — Piedmontese" lang="pms" hreflang="pms" data-title="Strop" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="گروہ (ریاضی) — Western Punjabi" lang="pnb" hreflang="pnb" data-title="گروہ (ریاضی)" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="artigo destacado"><a href="https://ro.wikipedia.org/wiki/Grup_(matematic%C4%83)" title="Grup (matematică) — romeno" lang="ro" hreflang="ro" data-title="Grup (matematică)" data-language-autonym="Română" data-language-local-name="romeno" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Группа (математика) — russo" lang="ru" hreflang="ru" data-title="Группа (математика)" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Gruppu_(matimatica)" title="Gruppu (matimatica) — siciliano" lang="scn" hreflang="scn" data-title="Gruppu (matimatica)" data-language-autonym="Sicilianu" data-language-local-name="siciliano" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) — servo-croata" lang="sh" hreflang="sh" data-title="Grupa (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="servo-croata" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Group_(mathematics)" title="Group (mathematics) — Simple English" lang="en-simple" hreflang="en-simple" data-title="Group (mathematics)" 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/Grupa_(matematika)" title="Grupa (matematika) — eslovaco" lang="sk" hreflang="sk" data-title="Grupa (matematika)" data-language-autonym="Slovenčina" data-language-local-name="eslovaco" 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/Grupa" title="Grupa — esloveno" lang="sl" hreflang="sl" data-title="Grupa" data-language-autonym="Slovenščina" data-language-local-name="esloveno" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Група (математика) — sérvio" lang="sr" hreflang="sr" data-title="Група (математика)" data-language-autonym="Српски / srpski" data-language-local-name="sérvio" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Grupp_(matematik)" title="Grupp (matematik) — sueco" lang="sv" hreflang="sv" data-title="Grupp (matematik)" data-language-autonym="Svenska" data-language-local-name="sueco" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Grupa_(matymatyka)" title="Grupa (matymatyka) — Silesian" lang="szl" hreflang="szl" data-title="Grupa (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="Silesian" class="interlanguage-link-target"><span>Ślůnski</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%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="குலம் (கணிதம்) — tâmil" lang="ta" hreflang="ta" data-title="குலம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tâmil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D3%AF%D2%B3_(%D1%80%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82)" title="Гурӯҳ (риёзиёт) — tajique" lang="tg" hreflang="tg" data-title="Гурӯҳ (риёзиёт)" data-language-autonym="Тоҷикӣ" data-language-local-name="tajique" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B8%E0%B8%9B_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="กรุป (คณิตศาสตร์) — tailandês" lang="th" hreflang="th" data-title="กรุป (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="tailandês" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Grupo_(matematika)" title="Grupo (matematika) — tagalo" lang="tl" hreflang="tl" data-title="Grupo (matematika)" data-language-autonym="Tagalog" data-language-local-name="tagalo" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Grup_(matematik)" title="Grup (matematik) — turco" lang="tr" hreflang="tr" data-title="Grup (matematik)" data-language-autonym="Türkçe" data-language-local-name="turco" 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%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Група (математика) — ucraniano" lang="uk" hreflang="uk" data-title="Група (математика)" data-language-autonym="Українська" data-language-local-name="ucraniano" 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/%DA%AF%D8%B1%D9%88%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="گروہ (ریاضی) — urdu" lang="ur" hreflang="ur" data-title="گروہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="artigo destacado"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_(to%C3%A1n_h%E1%BB%8Dc)" title="Nhóm (toán học) — vietnamita" lang="vi" hreflang="vi" data-title="Nhóm (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamita" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Groep_(algebra)" title="Groep (algebra) — West Flemish" lang="vls" hreflang="vls" data-title="Groep (algebra)" data-language-autonym="West-Vlams" data-language-local-name="West Flemish" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-vo mw-list-item"><a href="https://vo.wikipedia.org/wiki/Grup" title="Grup — volapuque" lang="vo" hreflang="vo" data-title="Grup" data-language-autonym="Volapük" 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.ombox-style{border:1px solid #fc3}.mw-parser-output .ombox-move{border:1px solid #9932cc}.mw-parser-output .ombox-protection{border:2px solid #a2a9b1}.mw-parser-output .ombox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .ombox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .ombox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .ombox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ombox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .ombox{margin:4px 10%}.mw-parser-output .ombox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}body.skin--responsive .mw-parser-output table.ombox img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .ombox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .ombox-speedy{background-color:#310402}}.mw-parser-output .tmbox{margin:4px 0;border-collapse:collapse;border:1px solid #c0c090;background-color:#f8eaba;box-sizing:border-box}.mw-parser-output .tmbox.mbox-small{font-size:88%;line-height:1.25em}.mw-parser-output .tmbox-speedy{border:2px solid #b32424;background-color:#fee7e6}.mw-parser-output .tmbox-delete{border:2px solid #b32424}.mw-parser-output .tmbox-content{border:1px solid #c0c090}.mw-parser-output .tmbox-style{border:2px solid #fc3}.mw-parser-output .tmbox-move{border:2px solid #9932cc}.mw-parser-output .tmbox .mbox-text{border:none;padding:0.25em 0.9em;width:100%}.mw-parser-output .tmbox .mbox-image{border:none;padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .tmbox .mbox-imageright{border:none;padding:2px 0.9em 2px 0;text-align:center}.mw-parser-output .tmbox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .tmbox .mbox-invalid-type{text-align:center}@media(min-width:720px){.mw-parser-output .tmbox{margin:4px 10%}.mw-parser-output .tmbox.mbox-small{clear:right;float:right;margin:4px 0 4px 1em;width:238px}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-night .mw-parser-output .tmbox-speedy{background-color:#310402}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmbox{background-color:#2e2505}html.skin-theme-clientpref-os .mw-parser-output .tmbox-speedy{background-color:#310402}}body.skin--responsive .mw-parser-output table.tmbox img{max-width:none!important}</style><table class="box-Sem_fontes plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div style="width:52px"><span typeof="mw:File"><a href="/wiki/Ficheiro:Question_book.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Question_book.svg/40px-Question_book.svg.png" decoding="async" width="40" height="32" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Question_book.svg/60px-Question_book.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Question_book.svg/80px-Question_book.svg.png 2x" data-file-width="252" data-file-height="199" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">Este artigo <b>não cita <a href="/wiki/Wikip%C3%A9dia:Verificabilidade" title="Wikipédia:Verificabilidade">fontes confiáveis</a></b>.<span class="hide-when-compact"> Ajude a <a href="/wiki/Wikip%C3%A9dia:Livro_de_estilo/Cite_as_fontes" title="Wikipédia:Livro de estilo/Cite as fontes">inserir referências</a>. Conteúdo não <a href="/wiki/Wikip%C3%A9dia:Verificabilidade" title="Wikipédia:Verificabilidade">verificável</a> pode ser removido.—<small><i>Encontre fontes:</i> <span class="plainlinks"><a rel="nofollow" class="external text" href="https://wikipedialibrary.wmflabs.org/">ABW</a>  •  <a rel="nofollow" class="external text" href="https://www.periodicos.capes.gov.br">CAPES</a>  •  <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&as_epq=Grupo+%28matem%C3%A1tica%29">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=Grupo+%28matem%C3%A1tica%29&oq=Grupo+%28matem%C3%A1tica%29">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=Grupo+%28matem%C3%A1tica%29">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=Grupo+%28matem%C3%A1tica%29">A</a>)</span></small></span> <small class="date-container"><i>(<span class="date">Dezembro de 2019</span>)</i></small></div></td></tr></tbody></table> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Ficheiro:Rubiks_revenge_tilt.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Rubiks_revenge_tilt.jpg/220px-Rubiks_revenge_tilt.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Rubiks_revenge_tilt.jpg/330px-Rubiks_revenge_tilt.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5a/Rubiks_revenge_tilt.jpg/440px-Rubiks_revenge_tilt.jpg 2x" data-file-width="600" data-file-height="600" /></a><figcaption>Os <a href="/wiki/Arranjo_(matem%C3%A1tica)" class="mw-redirect" title="Arranjo (matemática)">arranjos</a> possíveis do brinquedo <a href="/wiki/Cubo_de_Rubik#Variantes" title="Cubo de Rubik">A Vingança de Rubik</a> (versão 4x4x4 do Cubo de Rubik) formam um grupo.</figcaption></figure> <p>Em <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a>, um <b>grupo</b> é um <a href="/wiki/Conjunto" title="Conjunto">conjunto</a> de elementos associados a uma <a href="/wiki/Opera%C3%A7%C3%A3o_bin%C3%A1ria" title="Operação binária">operação</a> que combina dois elementos quaisquer para formar um terceiro. Para se qualificar como grupo o conjunto e a operação devem satisfazer algumas condições chamadas <a href="/wiki/Axioma" title="Axioma">axiomas</a> de grupo: <a href="/wiki/Associatividade" title="Associatividade">associatividade</a>, <a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a> e <a href="/wiki/Elemento_inverso" title="Elemento inverso">elementos inversos</a>. Apesar destes serem comuns a muitas estruturas matemáticas familiares - e.g. os <a href="/wiki/N%C3%BAmeros_inteiros" class="mw-redirect" title="Números inteiros">números inteiros</a> munidos da <a href="/wiki/Adi%C3%A7%C3%A3o" title="Adição">adição</a> formam um grupo - a formulação dos axiomas é independente da natureza concreta do grupo e sua operação. Isso permite lidar-se com entidade de origens matemáticas completamente diferentes de uma maneira flexível, mas retendo os aspectos estruturais essenciais de muitos objetos da <a href="/wiki/%C3%81lgebra_abstrata" title="Álgebra abstrata">álgebra abstrata</a> e além. A ubiquidade dos grupos em inúmeras áreas - dentro e fora da matemática - os tornam um princípio organizador central da matemática contemporânea. </p><p>Grupos compartilham um parentesco fundamental com a noção de <a href="/wiki/Simetria" title="Simetria">simetria</a>. Um <a href="/wiki/Grupo_de_simetria" title="Grupo de simetria">grupo de simetria</a> guarda informações sobre as simetrias de um objeto <a href="/wiki/Geometria" title="Geometria">geométrico</a>. Ele consiste do conjunto de transformações que preservam o objeto inalterado e a operação de <a href="/wiki/Composi%C3%A7%C3%A3o_de_fun%C3%A7%C3%B5es" title="Composição de funções">combinar</a> duas dessas transformações aplicando-as uma após a outra. Tais grupos de simetria, particularmente os <a href="/wiki/Grupo_de_Lie" title="Grupo de Lie">grupos de Lie</a> contínuos, têm um importante papel em muitas disciplinas. <a href="/wiki/Grupo_linear" class="mw-redirect" title="Grupo linear">Grupos de matrizes</a>, por exemplo, podem ser usados para compreender leis <a href="/wiki/F%C3%ADsica" title="Física">físicas</a> fundamentais da <a href="/wiki/Relatividade_especial" class="mw-redirect" title="Relatividade especial">relatividade especial</a> e fenômenos em <a href="/wiki/Qu%C3%ADmica" title="Química">química</a> <a href="/wiki/Mol%C3%A9cula" title="Molécula">molecular</a>. </p><p>O conceito de grupo emergiu do estudo de <a href="/wiki/Equa%C3%A7%C3%A3o_polinomial" title="Equação polinomial">equações de polinômios</a> com <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> na década de 1830. Após contribuições vindas de outros ramos da matemática, como <a href="/wiki/Teoria_dos_n%C3%BAmeros" title="Teoria dos números">teoria dos números</a> e geometria, a noção de grupo foi generalizada e se estabeleceu firmemente por volta de <a href="/wiki/1870" title="1870">1870</a>. A <a href="/wiki/Teoria_dos_grupos" title="Teoria dos grupos">teoria dos grupos</a> moderna - uma área muito ativa de pesquisa - estuda os grupos em si mesmos. Para explorá-los, matemáticos formularam várias noções para quebrar grupos em partes menores e mais compreensíveis, como <a href="/wiki/Subgrupo" title="Subgrupo">subgrupos</a>, <a href="/wiki/Grupo_quociente" title="Grupo quociente">grupos quocientes</a> e <a href="/wiki/Grupo_simples" title="Grupo simples">grupos simples</a>. Além das propriedades abstratas, matemáticos estudam as diferentes maneiras em que um grupo pode ser expresso concretamente (as <a href="/wiki/Representa%C3%A7%C3%A3o_de_grupo" title="Representação de grupo">representações</a> do grupo), tanto de um ponto-de-vista teorético quanto prático-computacional. Em particular, uma teoria ricamente desenvolvida é a dos <a href="/wiki/Grupo_finito" title="Grupo finito">grupos finitos</a>, que culminou com a monumental <a href="/wiki/Classifica%C3%A7%C3%A3o_dos_grupos_simples_finitos" title="Classificação dos grupos simples finitos">classificação dos grupos simples finitos</a>, completada em 1983. </p><p>Grupos estão por trás de muitas <a href="/wiki/Estrutura_alg%C3%A9brica" title="Estrutura algébrica">estruturas algébricas</a>, como <a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">corpos</a> e <a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">espaços vetoriais</a>, e são uma importante ferramenta para o estudo de simetrias. Por estas razões, a Teoria de Grupos é considerada uma área importante da matemática moderna, e tem muitas aplicações em Física Matemática, por exemplo em <a href="/wiki/F%C3%ADsica_de_part%C3%ADculas" title="Física de partículas">física de partículas</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definição"><span id="Defini.C3.A7.C3.A3o"></span>Definição</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=1" title="Editar secção: Definição" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=1" title="Editar código-fonte da secção: Definição"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seja G um conjunto e * uma <a href="/wiki/Opera%C3%A7%C3%A3o_bin%C3%A1ria" title="Operação binária">operação binária</a> definida sobre G. O <a href="/wiki/Par_ordenado" title="Par ordenado">par ordenado</a> (G,*) é um grupo se são satisfeitas os seguintes axiomas: </p> <ul><li><b>Fecho:</b> Para toda operação binária <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a*b=c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*b=c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2aa247b882375ed5430c67bc6f16014fbc44e06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.527ex; height:2.176ex;" alt="{\displaystyle a*b=c}"></span>, se <i>a</i> e <i>b</i> pertencem a G, <i>c</i> também deve pertencer a G</li> <li><b>Associatividade:</b> Quaisquer elementos <i>a,b,c</i> pertencentes a G, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a*b)*c=a*(b*c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∗</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a*b)*c=a*(b*c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5f454bc3f6f4cd80632e092956d92ce2afd5e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.964ex; height:2.843ex;" alt="{\displaystyle (a*b)*c=a*(b*c)}"></span></li> <li><b>Existência do <a href="/wiki/Elemento_neutro" title="Elemento neutro">elemento neutro</a>:</b> Existe um elemento <i><b>e</b></i> em G tal que <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 e*a=a*e=a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∗</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e*a=a*e=a,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa75cab8a7943aff2766c17b3e3e002a2cde6bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.09ex; height:2.009ex;" alt="{\displaystyle e*a=a*e=a,}"></span> para todo a pertencente a G.</li> <li><b>Existência do elemento simétrico:</b> Para qualquer elemento <i>a</i> em G, existe outro elemento <i>a' </i>em G, tal que, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a*a'=a'*a=e,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>a</mi> <mo>′</mo> </msup> <mo>∗</mo> <mi>a</mi> <mo>=</mo> <mi>e</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*a'=a'*a=e,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc93513af30b78aa301840b7a75afe207338cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.605ex; height:2.843ex;" alt="{\displaystyle a*a'=a'*a=e,}"></span> onde e é o elemento neutro previamente mencionado.</li></ul> <p>Apesar da relação estreita entre a operação "*" e a <a href="/wiki/Defini%C3%A7%C3%A3o" title="Definição">definição</a> de grupo, é possível denominar por grupo um conjunto G, desde que a operação em questão esteja evidente. </p><p>Ainda em relação à operação, os termos neutro e simétrico são frequentemente substituídos: </p> <table class="wikitable" cellspacing="1" cellpadding="1" border="1" style="width: 490px; height: 119px;"> <tbody><tr> <td>Operação </td> <td>Símbolo da operação </td> <td>Elemento neutro </td> <td>Elemento </td> <td>Simétrico de um elemento </td></tr> <tr> <td>Adição </td> <td style="text-align: center;">+ </td> <td>0 (<a href="/wiki/Zero" class="mw-redirect" title="Zero">Zero</a>) </td> <td>a </td> <td>-a (Oposto de a) </td></tr> <tr> <td><a href="/wiki/Multiplica%C3%A7%C3%A3o" title="Multiplicação">Multiplicação</a> </td> <td style="text-align: center;">. </td> <td>1 (Um) </td> <td>a </td> <td>a<sup>-1</sup> (Inverso de a) </td></tr> <tr> <td>Composição de funções </td> <td style="text-align: center;">o </td> <td>i(x)=x (identidade) </td> <td>a(x) </td> <td>a<sup>-1</sup>(x) (<a href="/wiki/Fun%C3%A7%C3%A3o_(matem%C3%A1tica)" title="Função (matemática)">Função</a> inversa de a(x)) </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Ordem">Ordem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=2" title="Editar secção: Ordem" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=2" title="Editar código-fonte da secção: Ordem"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A ordem de um grupo (G,*), onde G é finito, é o <a href="/wiki/Cardinalidade" title="Cardinalidade">número</a> de elementos do conjunto G. Caso G seja um conjunto <a href="/wiki/Infinito" title="Infinito">infinito</a>, dizemos que (G,*) tem ordem infinita. </p> <div class="mw-heading mw-heading2"><h2 id="Exemplos">Exemplos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=3" title="Editar secção: Exemplos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=3" title="Editar código-fonte da secção: Exemplos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>O menor grupo é formado por um único elemento.</li> <li>O conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,-1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,-1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab739c97a70285d651addcdd72021e5240b654c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{1,-1\}}"></span> é um grupo relativamente à multiplicação usual.</li> <li>O conjunto de todas as bijecções do conjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{1,2,\ldots ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{1,2,\ldots ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ebfec86b3f22a18f086275390917d5aaa2d8c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.257ex; height:2.843ex;" alt="{\displaystyle \{1,2,\ldots ,n\}}"></span> em si próprio é um grupo que se considerara como operação binária a <a href="/wiki/Composi%C3%A7%C3%A3o_de_fun%C3%A7%C3%B5es" title="Composição de funções">composição</a>. Este grupo representa-se por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a7e377e63c1f493aa1f3470f8af79e77c0c503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.29ex; height:2.509ex;" alt="{\displaystyle S_{n}.}"></span> Ver abaixo um detalhamento deste exemplo.</li> <li>Um exemplo de grupo de ordem finita é o grupo <a href="/wiki/Klein_4" title="Klein 4">Klein 4</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 G=\{e,a,b,c\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>e</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=\{e,a,b,c\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/989da531d98d8fdb46117bbab9e26b297dbfca02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.67ex; height:2.843ex;" alt="{\displaystyle G=\{e,a,b,c\}}"></span> onde <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> é o elemento neutro, todo elemento é seu próprio inverso, e as demais operações são definidas de forma que se <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feff4d40084c7351bf57b11ba2427f6331f5bdbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.977ex; height:2.009ex;" alt="{\displaystyle x,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> e <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> são três elementos distintos, então <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*y=z.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo>=</mo> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x*y=z.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d6e94cd3dd49a7cc207350f125cf242340d081" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.513ex; height:2.009ex;" alt="{\displaystyle x*y=z.}"></span></li> <li>O conjunto (<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} _{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7908e2fc1fba4549711182d0d8af65d2231441ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.416ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{n},}"></span> +), formado pelos números entre <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ea7e6006d94fc413924203051db13c3f6253a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.044ex; height:2.509ex;" alt="{\displaystyle n-1,}"></span> em que a soma é feita módulo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"></span> é um grupo. Por exemplo, em <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} _{42},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>42</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{42},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f595e574cc1fe4f0aa9e050d0f26dd643e3fbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.073ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} _{42},}"></span> temos que <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 20+30=50\mod 42=8.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>20</mn> <mo>+</mo> <mn>30</mn> <mo>=</mo> <mn>50</mn> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>42</mn> <mo>=</mo> <mn>8.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 20+30=50\mod 42=8.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5efa2b462135a9f566a7770c371f68619727761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:28.408ex; height:2.343ex;" alt="{\displaystyle 20+30=50\mod 42=8.}"></span></li> <li>O grupo de simetrias de um <a href="/wiki/Pol%C3%ADgono_regular" title="Polígono regular">polígono regular</a> de <i>n</i> lados, chamado <i>D<sub>n</sub></i> ou <a href="/wiki/Grupo_diedral" title="Grupo diedral">grupo diedral</a>. Ver abaixo um detalhamento deste exemplo.</li> <li>O conjunto <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_{n}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{n}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3260737a99bf9c22cd56e77a02a9dbe62b8c1609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.96ex; height:2.843ex;" alt="{\displaystyle M_{n}(\mathbb {R} )}"></span> das matrizes quadradas de ordem <i>n</i> sobre <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 '''R'''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mo>‴</mo> </msup> <msup> <mi>R</mi> <mo>‴</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle '''R'''}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44dc635ae4eb6056e7bda17ab6d8b2b9f3291641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.943ex; height:2.509ex;" alt="{\displaystyle '''R'''}"></span> não forma um grupo sob a multiplicação de matrizes, uma vez que a matriz nula, por exemplo, não admite um inverso. No entanto o subconjunto <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 GL_{n}(\mathbb {R} )=\{M\in M_{n}(\mathbb {R} ):det(M)\neq 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>M</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mi>d</mi> <mi>e</mi> <mi>t</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL_{n}(\mathbb {R} )=\{M\in M_{n}(\mathbb {R} ):det(M)\neq 0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f83f5df188ce28af1512b50391e1e615bc74fd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.37ex; height:2.843ex;" alt="{\displaystyle GL_{n}(\mathbb {R} )=\{M\in M_{n}(\mathbb {R} ):det(M)\neq 0\}}"></span> é um grupo sob a multiplicação.</li> <li>O conjunto das matrizes da forma</li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3860cb062ab9afba797465673e3ef3f5fca6147b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:13.018ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}1&a&b\\0&1&c\\0&0&1\end{pmatrix}}}"></span></dd></dl> <p>onde a, b e c são número reais, forma grupo com a multiplicação usual de matrizes. Esse é o chamado Grupo de Heisenberg. </p> <div class="mw-heading mw-heading3"><h3 id="Grupo_de_simetrias_de_um_quadrado">Grupo de simetrias de um quadrado</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=4" title="Editar secção: Grupo de simetrias de um quadrado" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=4" title="Editar código-fonte da secção: Grupo de simetrias de um quadrado"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quando construímos polígonos regulares, podemos ordenar os seus vértices para formar uma espécie de referência. Seja um polígono regular de ordem n. Ao considerarmos apenas as diversas configurações que não alteram o formato do polígono - modificando, portanto, somente as posições de seus vértices - temos o conjunto diedral de ordem n (representado por D<sub>n</sub>). A seguir, as possíveis configurações de um <a href="/wiki/Quadrado" title="Quadrado">quadrado</a>: </p> <table class="wikitable" border="1" style="text-align:center; margin:0 auto .5em auto;"> <tbody><tr> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_id.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Group_D8_id.svg/140px-Group_D8_id.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Group_D8_id.svg/210px-Group_D8_id.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Group_D8_id.svg/280px-Group_D8_id.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> id (mantê-lo como está)</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_90.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Group_D8_90.svg/140px-Group_D8_90.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Group_D8_90.svg/210px-Group_D8_90.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Group_D8_90.svg/280px-Group_D8_90.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> r<sub>1</sub> (rotação de 90° à direita)</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_180.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Group_D8_180.svg/140px-Group_D8_180.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Group_D8_180.svg/210px-Group_D8_180.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Group_D8_180.svg/280px-Group_D8_180.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> r<sub>2</sub> (rotação de 180° à direita)</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_270.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Group_D8_270.svg/140px-Group_D8_270.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/Group_D8_270.svg/210px-Group_D8_270.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/Group_D8_270.svg/280px-Group_D8_270.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> r<sub>3</sub> (rotação de 270° à direita) </td></tr> <tr> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_fv.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Group_D8_fv.svg/140px-Group_D8_fv.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/47/Group_D8_fv.svg/210px-Group_D8_fv.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/47/Group_D8_fv.svg/280px-Group_D8_fv.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> f<sub>v</sub> (Reflexão Vertical)</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_fh.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Group_D8_fh.svg/140px-Group_D8_fh.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Group_D8_fh.svg/210px-Group_D8_fh.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Group_D8_fh.svg/280px-Group_D8_fh.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> f<sub>h</sub> (Reflexão Horizontal)</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_f13.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Group_D8_f13.svg/140px-Group_D8_f13.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Group_D8_f13.svg/210px-Group_D8_f13.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0c/Group_D8_f13.svg/280px-Group_D8_f13.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> f<sub>d</sub> (Reflexão Diagonal)</td> <td><span typeof="mw:File"><a href="/wiki/Ficheiro:Group_D8_f24.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Group_D8_f24.svg/140px-Group_D8_f24.svg.png" decoding="async" width="140" height="118" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Group_D8_f24.svg/210px-Group_D8_f24.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a1/Group_D8_f24.svg/280px-Group_D8_f24.svg.png 2x" data-file-width="160" data-file-height="135" /></a></span><br /> f<sub>c</sub> (Reflexão Contra-Diagonal) </td></tr> <tr> <td style="text-align:left" colspan="4">As possíveis configurações (ou movimentos) obtidas a partir de rotações ou reflexões de um quadrado é denominado D<sub>4</sub>. </td></tr></tbody></table> <table class="wikitable" border="1" style="float:right; text-align:center; margin:.5em 0 .5em 1em; width:40ex; height:40ex;"> <caption><a href="/w/index.php?title=T%C3%A1bua_de_Cayley&action=edit&redlink=1" class="new" title="Tábua de Cayley (página não existe)">Tábua de Cayley</a> de D<sub>4</sub> </caption> <tbody><tr> <th width="12%" style="background:#FDD; border-top:solid black 2px; border-left:solid black 2px;">• </th> <th style="background:#FDD; border-top:solid black 2px;" width="11%">id </th> <th style="background:#FDD; border-top:solid black 2px;" width="11%">r<sub>1</sub> </th> <th style="background:#FDD; border-top:solid black 2px;" width="11%">r<sub>2</sub> </th> <th style="background:#FDD; border-right:solid black 2px; border-top:solid black 2px;" width="11%">r<sub>3</sub> </th> <th width="11%">f<sub>v</sub></th> <th width="11%">f<sub>h</sub></th> <th width="11%">f<sub>d</sub></th> <th width="11%">f<sub>c</sub> </th></tr> <tr> <th style="background:#FDD; border-left:solid black 2px;">id </th> <td style="background:#FDD;">id </td> <td style="background:#FDD;">r<sub>1</sub> </td> <td style="background:#FDD;">r<sub>2</sub> </td> <td style="background:#FDD; border-right:solid black 2px;">r<sub>3</sub></td> <td>f<sub>v</sub></td> <td>f<sub>h</sub></td> <td>f<sub>d</sub> </td> <td style="background:#FFFC93; border-right:solid black 2px; border-left:solid black 2px; border-top:solid black 2px;">f<sub>c</sub> </td></tr> <tr> <th style="background:#FDD; border-left:solid black 2px;">r<sub>1</sub> </th> <td style="background:#FDD;">r<sub>1</sub> </td> <td style="background:#FDD;">r<sub>2</sub> </td> <td style="background:#FDD;">r<sub>3</sub> </td> <td style="background:#FDD; border-right:solid black 2px;">id</td> <td>f<sub>c</sub></td> <td>f<sub>d</sub></td> <td>f<sub>v</sub> </td> <td style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;">f<sub>h</sub> </td></tr> <tr style="height:10%"> <th style="background:#FDD; border-left:solid black 2px;">r<sub>2</sub> </th> <td style="background:#FDD;">r<sub>2</sub> </td> <td style="background:#FDD;">r<sub>3</sub> </td> <td style="background:#FDD;">id </td> <td style="background:#FDD; border-right:solid black 2px;">r<sub>1</sub></td> <td>f<sub>h</sub></td> <td>f<sub>v</sub></td> <td>f<sub>c</sub> </td> <td style="background:#FFFC93; border-right: solid black 2px; border-left: solid black 2px;">f<sub>d</sub> </td></tr> <tr style="height:10%"> <th style="background:#FDD; border-bottom:solid black 2px; border-left:solid black 2px;">r<sub>3</sub> </th> <td style="background:#FDD; border-bottom:solid black 2px;">r<sub>3</sub> </td> <td style="background:#FDD; border-bottom:solid black 2px;">id </td> <td style="background:#FDD; border-bottom:solid black 2px;">r<sub>1</sub> </td> <td style="background:#FDD; border-right:solid black 2px; border-bottom:solid black 2px;">r<sub>2</sub></td> <td>f<sub>d</sub></td> <td>f<sub>c</sub> </td> <td>f<sub>h</sub> </td> <td style="background:#FFFC93; border-right:solid black 2px; border-left:solid black 2px; border-bottom:solid black 2px;">f<sub>v</sub> </td></tr> <tr style="height:10%"> <th>f<sub>v</sub> </th> <td>f<sub>v</sub></td> <td>f<sub>d</sub></td> <td>f<sub>h</sub></td> <td>f<sub>c</sub></td> <td>id</td> <td>r<sub>2</sub></td> <td>r<sub>1</sub></td> <td>r<sub>3</sub> </td></tr> <tr style="height:10%"> <th>f<sub>h</sub> </th> <td>f<sub>h</sub></td> <td>f<sub>c</sub></td> <td>f<sub>v</sub></td> <td style="background:#DDF;border:solid black 2px;">f<sub>d</sub></td> <td>r<sub>2</sub></td> <td>id</td> <td>r<sub>3</sub></td> <td>r<sub>1</sub> </td></tr> <tr style="height:10%"> <th>f<sub>d</sub> </th> <td>f<sub>d</sub></td> <td>f<sub>h</sub></td> <td>f<sub>c</sub></td> <td>f<sub>v</sub></td> <td>r<sub>3</sub></td> <td>r<sub>1</sub></td> <td>id</td> <td>r<sub>2</sub> </td></tr> <tr style="height:10%"> <th>f<sub>c</sub> </th> <td style="background:#9DFF93; border-left: solid black 2px; border-bottom: solid black 2px; border-top: solid black 2px;">f<sub>c</sub> </td> <td style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;">f<sub>v</sub> </td> <td style="background:#9DFF93; border-bottom: solid black 2px; border-top: solid black 2px;">f<sub>d</sub> </td> <td style="background:#9DFF93; border-bottom:solid black 2px; border-top:solid black 2px; border-right:solid black 2px;">f<sub>h</sub></td> <td>r<sub>1</sub></td> <td>r<sub>3</sub></td> <td>r<sub>2</sub></td> <td>id </td></tr> <tr> <td colspan="9" style="text-align:left">Os elementos id, r<sub>1</sub>, r<sub>2</sub> e r<sub>3</sub> formam um subgrupo de D<sub>4</sub>, colorido em vermelho. Em verde e amarelo, classes laterais esquerda e direita desse subgrupo, respectivamente. </td></tr></tbody></table> <p>Estabelecendo a <a href="/wiki/Opera%C3%A7%C3%A3o" class="mw-disambig" title="Operação">operação</a> sobre este conjunto "<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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span>", definida por: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in D_{n},a*b=c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>=</mo> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in D_{n},a*b=c,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03228fac64d4c3d421530d9e27680d33ae904d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.453ex; height:2.509ex;" alt="{\displaystyle a,b\in D_{n},a*b=c,}"></span> onde <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"></span> é a configuração obtida após executar o movimento <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> e em seguida o movimento <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 b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef051eb30c89e5493d672f6479566c673b0890a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.644ex; height:2.176ex;" alt="{\displaystyle b.}"></span> </p><p>A partir da operação entre quaisquer elementos de D<sub>4</sub>, é possível verificar que o resultado também é um elemento de D<sub>4</sub>. Por exemplo, <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 r_{1}*r_{1}=r_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∗</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}*r_{1}=r_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e809e0cfdb4e3c0cf28401bcc79b432a3c3d6c11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.249ex; height:2.009ex;" alt="{\displaystyle r_{1}*r_{1}=r_{2},}"></span> ou <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 f_{v}*f_{v}=id.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> <mo>=</mo> <mi>i</mi> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{v}*f_{v}=id.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8def8e4ab6bb89e6448d6bdfb38397315d1dae7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.296ex; height:2.509ex;" alt="{\displaystyle f_{v}*f_{v}=id.}"></span> Como D<sub>4</sub> se trata de um <a href="/wiki/Conjunto_finito" title="Conjunto finito">conjunto finito</a>, é perfeitamente possível construir uma tabela com os resultados da operação entre quaisquer dois de seus elementos. </p><p>Com o auxílio de tal tabela, verificamos as seguintes propriedades de D<sub>4</sub> em relação à "<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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span>": </p> <ul><li>Para quaisquer elementos a, b e c de D<sub>4</sub>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a*b)*c=a*(b*c);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∗</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a*b)*c=a*(b*c);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b125940493572631b3686e588933171b28e41cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.611ex; height:2.843ex;" alt="{\displaystyle (a*b)*c=a*(b*c);}"></span></li> <li>Existe um elemento <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> que, operado a qualquer outro elemento y, resulta em y ou seja: <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 \exists e\in D_{4}:\forall y\in D_{4},e*y=y*e=y;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∃</mi> <mi>e</mi> <mo>∈</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>:</mo> <mi mathvariant="normal">∀</mi> <mi>y</mi> <mo>∈</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mi>e</mi> <mo>∗</mo> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mi>y</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exists e\in D_{4}:\forall y\in D_{4},e*y=y*e=y;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61eaf5f1445366c560e0f6d00083fa8ff60eeab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.3ex; height:2.509ex;" alt="{\displaystyle \exists e\in D_{4}:\forall y\in D_{4},e*y=y*e=y;}"></span></li> <li>Para todo elemento <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\in D_{4},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in D_{4},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2207413d4252d1ba8195bfeaf3a55d68a0ef27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.796ex; height:2.509ex;" alt="{\displaystyle x\in D_{4},}"></span> existe outro elemento <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'\in D_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>∈</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'\in D_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04005049233de3eb07a4b8c834689b4d4938b227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.834ex; height:2.843ex;" alt="{\displaystyle x'\in D_{4}}"></span> tal que, quando operados (não importando a ordem) resultam no elemento <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> do item anterior, ou seja: <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 \forall x\in D_{4},\exists x':x*x'=x'*x=e.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>∈</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <mi mathvariant="normal">∃</mi> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>:</mo> <mi>x</mi> <mo>∗</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>∗</mo> <mi>x</mi> <mo>=</mo> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall x\in D_{4},\exists x':x*x'=x'*x=e.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d30d69b1ab81b643d6abfb6f2ebdb2788b73f1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.724ex; height:2.843ex;" alt="{\displaystyle \forall x\in D_{4},\exists x':x*x'=x'*x=e.}"></span></li></ul> <p>Claramente, o elemento <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> em questão é <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 id,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>d</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle id,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59a0bc46bdace3a428247f7b60ca992016eddeea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.665ex; height:2.509ex;" alt="{\displaystyle id,}"></span> pois ao operá-lo a qualquer elemento, o mesmo não tem sua configuração alterada. A terceira propriedade é verificada nas linhas e colunas da tabela dos possíveis resultados da operação * em relação aos elementos de D<sub>4</sub>. Em cada linha e cada coluna, verificamos que o elemento <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 id}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle id}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/514c0ded05af41857b8728628c0a664c1b9b1b37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.018ex; height:2.176ex;" alt="{\displaystyle id}"></span> aparece uma única vez. Portanto, para qualquer elemento x, existe outro elemento x' que, operado ao primeiro, resulta em id. </p> <div class="mw-heading mw-heading3"><h3 id="Grupo_das_permutações"><span id="Grupo_das_permuta.C3.A7.C3.B5es"></span>Grupo das permutações</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=5" title="Editar secção: Grupo das permutações" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=5" title="Editar código-fonte da secção: Grupo das permutações"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Seja o conjunto <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 U=\{1,2,\ldots ,n\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\{1,2,\ldots ,n\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff35e3cc8a48b17cc1930ae4f4881213a89066df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.785ex; height:2.843ex;" alt="{\displaystyle U=\{1,2,\ldots ,n\},}"></span> uma <a href="/wiki/Permuta%C3%A7%C3%A3o" title="Permutação">permutação</a> em U, é uma função <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 f:U\rightarrow U,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>U</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:U\rightarrow U,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac90f8334ab9b9df0cad4b95d9a46a5f24815c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.042ex; height:2.509ex;" alt="{\displaystyle f:U\rightarrow U,}"></span> tal que f é <a href="/wiki/Bijetora" class="mw-redirect" title="Bijetora">bijetora</a>. O conjunto de todas as permutações em U é chamado de conjunto das permutações de n elementos. Uma permutação pode ser representada de forma matricial, onde <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 f={\begin{pmatrix}1&2&3&4\\1&3&2&4\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\begin{pmatrix}1&2&3&4\\1&3&2&4\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b294f9276b50d40fe23f1f28f0b8895a038e448" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.167ex; height:6.176ex;" alt="{\displaystyle f={\begin{pmatrix}1&2&3&4\\1&3&2&4\end{pmatrix}}}"></span> significa que f(1)=1, f(2)=3, f(3)=2 e f(4)=4. </p> <table class="wikitable" width="200" cellspacing="1" cellpadding="1" border="1" align="right"> <caption><a href="/w/index.php?title=T%C3%A1bua_de_Cayley&action=edit&redlink=1" class="new" title="Tábua de Cayley (página não existe)">Tábua de Cayley</a> de S<sub>3</sub> </caption> <tbody><tr> <td style="text-align: center;"><span style="font-weight: bold;">•</span> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span></b> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span></b> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span></b> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span></b> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span></b> </td> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span></b> </td></tr> <tr> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span> </td></tr> <tr> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span> </td></tr> <tr> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span> </td></tr> <tr> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> </td></tr> <tr> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> </td></tr> <tr> <td><b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span></b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5f49ca9f3cde19f364ea96c7d35fa36812b633" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{6}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48763cb9875de7b4125c40e95e542e775c1cbc4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{4}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9446d54bf4a29799ff14169ffc7ba4423102fd08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{5}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87858df7457aa93caaef5a316db87a7240cc8c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{2}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92db8e65bb75b79799f0f3a29e975b37e227069" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{3}}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span> </td></tr> <tr> <td colspan="9" style="text-align:left">Tábua do grupo S<sub>3</sub> </td></tr></tbody></table> <p>Assim, o conjunto das permutações de n elementos, para n=3, consiste nos elementos: </p> <table width="200" cellspacing="1" cellpadding="1" border="0"> <tbody><tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}={\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}={\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86859a010ef1a6b4261e129380fec538da73cbb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.597ex; height:6.176ex;" alt="{\displaystyle P_{1}={\begin{pmatrix}1&2&3\\1&2&3\end{pmatrix}};}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{2}={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{2}={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d858dcd02f692ad4cdc6c5b872adc10d8e314c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.597ex; height:6.176ex;" alt="{\displaystyle P_{2}={\begin{pmatrix}1&2&3\\2&3&1\end{pmatrix}};}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{3}={\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{3}={\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47f04a8333f174b1cd96540bcd6f5aca6aa56564" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.597ex; height:6.176ex;" alt="{\displaystyle P_{3}={\begin{pmatrix}1&2&3\\3&1&2\end{pmatrix}};}"></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{4}={\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{4}={\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/439051a6dfd7b03edc023e9a19065dc145ad6f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.597ex; height:6.176ex;" alt="{\displaystyle P_{4}={\begin{pmatrix}1&2&3\\1&3&2\end{pmatrix}};}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{5}={\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{5}={\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5816cbd1eded563f954912f0f1f35ffee4e84e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.597ex; height:6.176ex;" alt="{\displaystyle P_{5}={\begin{pmatrix}1&2&3\\3&2&1\end{pmatrix}};}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{6}={\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{6}={\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0d8accdb7affd7efca1d79b6a7353c04dca8c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.95ex; height:6.176ex;" alt="{\displaystyle P_{6}={\begin{pmatrix}1&2&3\\2&1&3\end{pmatrix}}}"></span> </td></tr></tbody></table> <p>Considerando o conjunto descrito acima e a composição de funções, temos um par ordenado que satisfaz as propriedades de um grupo, pois, a composição de funções é sempre <i>associativa</i> , existe um <i>elemento neutro</i> (no caso, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/398f438d75434e6fbf48dc232c1ad7228a738568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{1}}"></span>) e todas as funções são bijetoras, e portanto, inversíveis (todos os elementos possuem um simétrico). O mesmo argumento pode ser usado para provar que, para qualquer n positivo, o conjunto das permutações de n elementos forma grupo, em relação à composição de funções. Esse grupo é denominado <i>grupo simétrico de n elementos</i>, e é representado por <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5a7e377e63c1f493aa1f3470f8af79e77c0c503" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.29ex; height:2.509ex;" alt="{\displaystyle S_{n}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Propriedades_Imediatas">Propriedades Imediatas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=6" title="Editar secção: Propriedades Imediatas" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=6" title="Editar código-fonte da secção: Propriedades Imediatas"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>A identidade de um grupo é única.</b> <i>Demonstração:</i> suponha <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> e <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 e'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c198f6710d781baaf94653df305a4881380033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.768ex; height:2.509ex;" alt="{\displaystyle e'}"></span> são duas identidades. Então, para todo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}"></span> é verdade que <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*e'=e'*g=g.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∗</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g*e'=e'*g=g.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e17532b6df0c85074b79ca5251377c48b175de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.118ex; height:2.843ex;" alt="{\displaystyle g*e'=e'*g=g.}"></span> Em particular, temos <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 e*e'=e.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∗</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e*e'=e.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48b0899a21b225a00873b2f6619d9427ebd9958" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.875ex; height:2.509ex;" alt="{\displaystyle e*e'=e.}"></span> Também é verdade que, para todo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a2c972dfcbb2bb5f88ddfd1b997e0a08c21363" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.474ex; height:2.509ex;" alt="{\displaystyle G,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*e=e*g=g.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mi>e</mi> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g*e=e*g=g.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34396b04c3e1eee5485532f230eeb334e323af24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.748ex; height:2.009ex;" alt="{\displaystyle g*e=e*g=g.}"></span> Em particular, para <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=e',}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=e',}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3d971b9cc2c940b94b3fdb9f89160334c01dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.63ex; height:2.843ex;" alt="{\displaystyle g=e',}"></span> temos <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 e*e'=e'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∗</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e*e'=e'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec8ff73dce854d178441f0448656bd5cb0c6caa7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.56ex; height:2.509ex;" alt="{\displaystyle e*e'=e'.}"></span> Portanto, <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 e=e*e'=e'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mi>e</mi> <mo>∗</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>e</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e=e*e'=e'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63c875d6a295b62ede1cff59c9fba2f5cbf02329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.742ex; height:2.509ex;" alt="{\displaystyle e=e*e'=e'.}"></span> Note-se que esta prova não usa nenhuma outra propriedade do grupo além da existência da identidade. </p><p><b>Um elemento de um grupo G possui apenas um inverso.</b> <i>Demonstração:</i> seja <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> ∈ <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> e sejam <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac74959896052e160a5953102e4bc3850fe93b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.014ex; height:2.509ex;" alt="{\displaystyle x'}"></span> inversos de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23a3f421f58ef3bc6f9ec70e883e1496ff871e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.763ex; height:2.009ex;" alt="{\displaystyle g.}"></span> Então </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 x=x*e=x*(g*x')=(x*g)*x'=e*x'=x'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mi>x</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>e</mi> <mo>∗</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=x*e=x*(g*x')=(x*g)*x'=e*x'=x'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14ef2b14b8b25dbf3d88b0f4156468919e71bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.701ex; height:3.009ex;" alt="{\displaystyle x=x*e=x*(g*x')=(x*g)*x'=e*x'=x'.}"></span></dd></dl> <p>Está visto que o elemento inverso de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> é único. Representa-se por <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^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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 g^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/266be6d2c79095921dab152e047dab7f0d7f8e65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.098ex; height:3.009ex;" alt="{\displaystyle g^{-1}.}"></span> </p><p><b>Em um grupo temos <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 (xy)^{-1}=y^{-1}x^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)^{-1}=y^{-1}x^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b459c36b81a36ccec9b1a4cfe95987603ccaa235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.528ex; height:3.176ex;" alt="{\displaystyle (xy)^{-1}=y^{-1}x^{-1}.}"></span> </b> <i>Demonstração:</i> Temos que <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 (xy)^{-1}(xy)=e.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>e</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)^{-1}(xy)=e.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a844d36f5edac136baa083ed710ac1eee3a6548d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.751ex; height:3.176ex;" alt="{\displaystyle (xy)^{-1}(xy)=e.}"></span> Aplicando <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 y^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8e104015ae3c898ae49c8df5de22d1694e07de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:3.009ex;" alt="{\displaystyle y^{-1}}"></span> nos dois lados da igualdade temos: <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 (xy)^{-1}(xy)(y^{-1})=e(y^{-1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>e</mi> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)^{-1}(xy)(y^{-1})=e(y^{-1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8806605c921d8be54893fbd0aa465e49914998e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.356ex; height:3.176ex;" alt="{\displaystyle (xy)^{-1}(xy)(y^{-1})=e(y^{-1}).}"></span> Pela associatividade e definição de elemento neutro temos: <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 (xy)^{-1}x=y^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)^{-1}x=y^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc32b5c43e061dcaf0f47e1c7c7c07ed0756f934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.196ex; height:3.176ex;" alt="{\displaystyle (xy)^{-1}x=y^{-1}.}"></span> Repetindo o procedimento para <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf91609f1a0b7847e108023b015cb6b0d567821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.662ex; height:2.676ex;" alt="{\displaystyle x^{-1}}"></span> no lugar de <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 y^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb8e104015ae3c898ae49c8df5de22d1694e07de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.493ex; height:3.009ex;" alt="{\displaystyle y^{-1}}"></span> finalmente obtemos <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 (xy)^{-1}=y^{-1}x^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)^{-1}=y^{-1}x^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b459c36b81a36ccec9b1a4cfe95987603ccaa235" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.528ex; height:3.176ex;" alt="{\displaystyle (xy)^{-1}=y^{-1}x^{-1}.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Conceitos_básicos"><span id="Conceitos_b.C3.A1sicos"></span>Conceitos básicos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=7" title="Editar secção: Conceitos básicos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=7" title="Editar código-fonte da secção: Conceitos básicos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A exemplo de diversas outras estruturas, é conveniente para o estudo de grupos a definição de homomorfismos subgrupos e quocientes. </p> <div class="mw-heading mw-heading3"><h3 id="Homomorfismos_de_grupos">Homomorfismos de grupos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=8" title="Editar secção: Homomorfismos de grupos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=8" title="Editar código-fonte da secção: Homomorfismos de grupos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sejam <i>(G,*)</i> e <i>(H,*)</i> dois grupos e <i>f</i> uma função de <i>G</i> em <i>H</i>, então dizemos que <i>f</i> é um homomorfismo se </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 (\forall x,y\in G):f(x*y)=f(x)*f(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall x,y\in G):f(x*y)=f(x)*f(y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c715f56f689a9994fc139dd69906abe70c67ae50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.594ex; height:2.843ex;" alt="{\displaystyle (\forall x,y\in G):f(x*y)=f(x)*f(y).}"></span></dd></dl> <p>Em outras palavras, a função <i>f</i> preserva a operação do grupo G. Se a função se trata de uma bijeção ela é chamada de isomorfismo e os grupos <i>G</i> e <i>H</i> são ditos isomorfos. </p> <div class="mw-heading mw-heading3"><h3 id="Exemplos_2">Exemplos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=9" title="Editar secção: Exemplos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=9" title="Editar código-fonte da secção: Exemplos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Para qualquer grupo <i>G</i>, a função <i>f</i> de <i>G</i> em <i>G</i> definida por <i>f(x)=x</i> é um homomorfismo. O mesmo acontece se se definirmos <i>f(x)=e</i>, onde <i>e</i> é a identidade de <i>G</i>.</li> <li>Considere os grupos <b>R</b> \ <i>{0}</i> e <i>{1,-1}</i> ambos com a multiplicação usual. Então a função <i>f</i> de <b>R</b> \ <i>{0}</i> em <i>{1,-1}</i> definida por <i>f(x)=x/|x|</i> é um <a href="/wiki/Homomorfismo_de_grupos" title="Homomorfismo de grupos">homomorfismo de grupos</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Propriedades">Propriedades</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=10" title="Editar secção: Propriedades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=10" title="Editar código-fonte da secção: Propriedades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <i>f</i> for um homomorfismo de <i>G</i> em <i>H</i> e se <i>e<sub>G</sub></i> e <i>e<sub>H</sub></i> forem os elementos neutros de <i>G</i> e de <i>H</i> respectivamente, então <i>f(e<sub>G</sub>)=e<sub>H</sub></i>. Isto porque </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 f(e_{G})=f(e_{G}*e_{G})=f(e_{G})*f(e_{G})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(e_{G})=f(e_{G}*e_{G})=f(e_{G})*f(e_{G})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29dd5ec58b8617196b20d4c82356e7341b7630a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.975ex; height:2.843ex;" alt="{\displaystyle f(e_{G})=f(e_{G}*e_{G})=f(e_{G})*f(e_{G})}"></span></dd></dl> <p>e e<sub>H</sub> é o único elemento <i>x</i> ∈ <i>H</i> tal que <i>x*x=x</i>. </p><p>Se <i>f</i> for um homomorfismo de <i>G</i> em <i>H</i> e se <i>x</i> ∈ <i>G</i>, então </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 f(x)^{-1}=f(x^{-1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)^{-1}=f(x^{-1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29efeb9482c5bda559fccd253555b941c57fd92e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.246ex; height:3.176ex;" alt="{\displaystyle f(x)^{-1}=f(x^{-1}).}"></span></dd></dl> <p>Isto porque </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 f(x^{-1})*f(x)=f(x^{-1}*x)=f(e_{G})=e_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∗</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x^{-1})*f(x)=f(x^{-1}*x)=f(e_{G})=e_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45ebf6aa7a956ad0e8d50967558c33529fc46e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.403ex; height:3.176ex;" alt="{\displaystyle f(x^{-1})*f(x)=f(x^{-1}*x)=f(e_{G})=e_{H}}"></span></dd></dl> <p>e, portanto, <i>f(x<sup>−1</sup>)</i> é o inverso de <i>f(x)</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Tipos_de_homomorfismos">Tipos de homomorfismos</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=11" title="Editar secção: Tipos de homomorfismos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=11" title="Editar código-fonte da secção: Tipos de homomorfismos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Se <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> for um homomorfismo de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef601e1519093ba6c2944b945882c119f990e704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.71ex; height:2.509ex;" alt="{\displaystyle H,}"></span> diz-se que </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> é um <i>monomorfismo</i> se for injectivo;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> é um <i>epimorfismo</i> se for sobrejectivo;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> é um <i>isomorfismo</i> se for simultaneamente um monomorfismo e um epimorfismo, ou seja, se for uma bijecção;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> é um <i>endomorfismo</i> se <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=H;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>=</mo> <mi>H</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G=H;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dca7e70da0c4fe7bca7e3a45bb173e32cd0419d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.636ex; height:2.509ex;" alt="{\displaystyle G=H;}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> é um <i><a href="/wiki/Automorfismo" title="Automorfismo">automorfismo</a></i> se for simultaneamente um endomorfismo e um isomorfismo.</li></ul> <p>Se <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> for um isomorfismo, então tem uma inversa (pois é uma bijecção). A função <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 f^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}"></span> é também um homomorfismo de grupos e, portanto, um isomorfismo. </p><p>Diz-se que dois grupos <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> e <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> são <i>isomorfos</i> se existir um isomorfismo de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc645a5b7e8a2022ad70fc42dbda04c008a33a9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.474ex; height:2.176ex;" alt="{\displaystyle G.}"></span> </p><p>Exemplos: </p> <ul><li>O grupo <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 (}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ccb001e791b6a7340825e0e0b1a60e4e03f3d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.905ex; height:2.843ex;" alt="{\displaystyle (}"></span><b>R</b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7d2e158decdc78ebbc1447ac68698998e9f136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.747ex; height:2.843ex;" alt="{\displaystyle ,+)}"></span> dos números reais (com a adição) e o grupo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (]0,+\infty [,.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">[</mo> <mo>,</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (]0,+\infty [,.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47d99a6896978b87f46eaa58671351521b34117a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.499ex; height:2.843ex;" alt="{\displaystyle (]0,+\infty [,.)}"></span> dos números reais maiores do que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> (com a multiplicação) são isomorfos, pois a <a href="/wiki/Fun%C3%A7%C3%A3o_exponencial" title="Função exponencial">função exponencial</a> é um isomorfismo de <b>R</b> em <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ]0,+\infty [.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">[</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ]0,+\infty [.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df3e3e4819e891ae4e93a04e0f1a91ff9696e7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.269ex; height:2.843ex;" alt="{\displaystyle ]0,+\infty [.}"></span></li> <li>O grupo <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 (}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ccb001e791b6a7340825e0e0b1a60e4e03f3d78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:0.905ex; height:2.843ex;" alt="{\displaystyle (}"></span><b>Q</b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ,+)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7d2e158decdc78ebbc1447ac68698998e9f136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.747ex; height:2.843ex;" alt="{\displaystyle ,+)}"></span> dos números racionais (com a adição) e o grupo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (]0,+\infty [}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">]</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">[</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (]0,+\infty [}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab8b8b2a337eee0d3308fdb6823dbafb45d93e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.527ex; height:2.843ex;" alt="{\displaystyle (]0,+\infty [}"></span> ∩ <b>Q</b><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,.)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>,</mo> <mo>.</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ,.)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6f439955b0cd658c68b61cb6fe1268661c520b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.973ex; height:2.843ex;" alt="{\displaystyle ,.)}"></span> dos números racionais maiores do que <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"></span> (com a multiplicação) não são isomorfos. Basta ver que se <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> for um homomorfismo do primeiro no segundo e que se houver algum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> ∈ <b>Q</b> tal que <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 f(q)=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(q)=2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b02b119e496baf980144054579db704397b632ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.065ex; height:2.843ex;" alt="{\displaystyle f(q)=2,}"></span> então</li></ul> <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 f(q/2)^{2}=f(q/2).f(q/2)=f(q/2+q/2)=f(q)=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(q/2)^{2}=f(q/2).f(q/2)=f(q/2+q/2)=f(q)=2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8a0db661e52b0f5ad7170b46c5af21d9b0a45bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.613ex; height:3.176ex;" alt="{\displaystyle f(q/2)^{2}=f(q/2).f(q/2)=f(q/2+q/2)=f(q)=2,}"></span></dd></dl> <p>mas <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> não tem nenhuma raiz quadrada racional. </p> <div class="mw-heading mw-heading2"><h2 id="Subgrupos">Subgrupos</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=12" title="Editar secção: Subgrupos" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=12" title="Editar código-fonte da secção: Subgrupos"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="hatnote"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/17px-Magnifying_glass_01.svg.png" decoding="async" width="17" height="17" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/26px-Magnifying_glass_01.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3a/Magnifying_glass_01.svg/34px-Magnifying_glass_01.svg.png 2x" data-file-width="663" data-file-height="659" /></span></span>Ver artigo principal: <a href="/wiki/Subgrupo" title="Subgrupo">Subgrupo</a></div> <p><b>Definição:</b> Dado um grupo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (G,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (G,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e54a87abf331634c8962ef14c4c5ec41f94fd29c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.832ex; height:2.843ex;" alt="{\displaystyle (G,*)}"></span> dizemos que um subconjunto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> é um subgrupo, quando <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (H,*)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>H</mi> <mo>,</mo> <mo>∗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (H,*)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ca9efa8f60b1acd949386d15ff163c637042933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.069ex; height:2.843ex;" alt="{\displaystyle (H,*)}"></span> é um grupo. </p> <div class="mw-heading mw-heading2"><h2 id="Ver_também"><span id="Ver_tamb.C3.A9m"></span>Ver também</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&veaction=edit&section=13" title="Editar secção: Ver também" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grupo_(matem%C3%A1tica)&action=edit&section=13" title="Editar código-fonte da secção: Ver também"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="infobox noprint" style="width:250px; line-height:2.2em; font-size:90%"> <tbody><tr style="line-height:1.3em"> <td colspan="2" style="text-align: center;">Outros projetos <a href="/wiki/Wikimedia" class="mw-redirect" title="Wikimedia">Wikimedia</a> também contêm material sobre este tema: </td></tr> <tr> <th><span typeof="mw:File"><a href="https://pt.wikibooks.org/wiki/Special:Search/%C3%81lgebra_abstrata/Mon%C3%B3ides_e_grupos" title="Wikilivros"><img alt="Wikilivros" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/21px-Wikibooks-logo.svg.png" decoding="async" width="21" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/42px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> </th> <td><a href="https://pt.wikibooks.org/wiki/Special:Search/%C3%81lgebra_abstrata/Mon%C3%B3ides_e_grupos" class="extiw" title="b:Special:Search/Álgebra abstrata/Monóides e grupos"><span title="Procurar por livros e manuais no Wikilivros"><b>Livros e manuais</b></span></a> no <a href="https://pt.wikibooks.org/wiki/P%C3%A1gina_principal" class="extiw" title="b:Página principal"><span title="Wikilivros">Wikilivros</span></a> </td></tr> </tbody></table><div id="interProject" style="display:none;"> <ul><li><a href="https://pt.wikibooks.org/wiki/Special:Search/%C3%81lgebra_abstrata/Mon%C3%B3ides_e_grupos" class="extiw" title="b:Special:Search/Álgebra abstrata/Monóides e grupos"><span title="Wikilivros">Wikilivros</span></a></li></ul> </div> <ul><li><a href="/wiki/Ac%C3%A7%C3%A3o_de_um_grupo" class="mw-redirect" title="Acção de um grupo">Acção de um grupo</a></li> <li><a href="/wiki/Grup%C3%B3ide_(estrutura_alg%C3%A9brica)" class="mw-redirect" title="Grupóide (estrutura algébrica)">Grupóide (estrutura algébrica)</a>, apenas um conjunto com uma operação binária</li> <li><a href="/wiki/Mon%C3%B3ide" class="mw-redirect" title="Monóide">Monóide</a>, quando a operação binária é <a href="/wiki/Associatividade" title="Associatividade">associativa</a> e tem elemento neutro, mas não necessariamente tem elemento inverso</li> <li><a href="/wiki/Semigrupo" title="Semigrupo">Semigrupo</a></li> <li><a href="/wiki/Grupo_topol%C3%B3gico" title="Grupo topológico">Grupo topológico</a>, quando existe uma topologia consistente com a operação binária</li> <li><a href="/wiki/Grupo_abeliano" title="Grupo abeliano">Grupo abeliano</a>, um grupo em que a operação binária é <a href="/wiki/Comutatividade" title="Comutatividade">comutativa</a></li> <li><a href="/wiki/Grupo_ordenado" title="Grupo ordenado">Grupo ordenado</a>, um grupo com uma <a href="/wiki/Rela%C3%A7%C3%A3o_de_ordem" title="Relação de ordem">relação de ordem</a> compatível com a sua <a href="/wiki/Opera%C3%A7%C3%A3o_bin%C3%A1ria" title="Operação binária">operação binária</a></li> <li><a href="/wiki/Grupo_de_simetria" title="Grupo de simetria">Grupo de simetria</a></li> <li><a href="/wiki/Grupo_diedral" title="Grupo diedral">Grupo diedral</a></li></ul> <p><br /> </p> <div role="navigation" class="navbox" aria-labelledby="Tópicos_principais_sobre_álgebra" style="padding:3px"><table class="nowraplinks hlist collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div class="plainlinks hlist navbar mini"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:%C3%81lgebra" title="Predefinição:Álgebra"><abbr title="Ver esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">v</abbr></a></li><li class="nv-discutir"><a href="/w/index.php?title=Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:%C3%81lgebra&action=edit&redlink=1" class="new" title="Predefinição Discussão:Álgebra (página não existe)"><abbr title="Discutir esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">d</abbr></a></li><li class="nv-editar"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Predefini%C3%A7%C3%A3o:%C3%81lgebra&action=edit"><abbr title="Editar esta predefinição" style=";;background:none transparent;border:none;-moz-box-shadow:none;-webkit-box-shadow:none;box-shadow:none; padding:0;">e</abbr></a></li></ul></div><div id="Tópicos_principais_sobre_álgebra" style="font-size:114%;margin:0 4em">Tópicos principais sobre <a href="/wiki/%C3%81lgebra" title="Álgebra">álgebra</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Geral</th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%C3%81lgebra_elementar" title="Álgebra elementar">Álgebra elementar</a></li> <li><a href="/wiki/%C3%81lgebra_abstrata" title="Álgebra abstrata">Álgebra abstrata</a></li> <li><a href="/wiki/%C3%81lgebra_comutativa" title="Álgebra comutativa">Álgebra comutativa</a></li> <li><a href="/wiki/Teoria_da_ordem" title="Teoria da ordem">Teoria da ordem</a></li> <li><a href="/wiki/Teoria_das_categorias" title="Teoria das categorias">Teoria das categorias</a></li> <li><a href="/wiki/K-Teoria_(matem%C3%A1tica)" class="mw-redirect" title="K-Teoria (matemática)">K-Teoria</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Estrutura_alg%C3%A9brica" title="Estrutura algébrica">Estruturas algébricas</a></th><td class="navbox-list navbox-even" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a class="mw-selflink selflink">Grupo</a> – <a href="/wiki/Teoria_dos_grupos" title="Teoria dos grupos">Teoria dos grupos</a></li> <li><a href="/wiki/Anel_(matem%C3%A1tica)" title="Anel (matemática)">Anel</a> – <a href="/wiki/Teoria_dos_an%C3%A9is" title="Teoria dos anéis">Teoria dos anéis</a></li> <li><a href="/wiki/Corpo_(matem%C3%A1tica)" title="Corpo (matemática)">Corpo</a> – <a href="/wiki/Teoria_dos_corpos" title="Teoria dos corpos">Teoria dos corpos</a></li> <li><a href="/wiki/%C3%81lgebra_universal" title="Álgebra universal">Álgebra universal</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%C3%81lgebra_linear" title="Álgebra linear">Álgebra linear</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Matriz_(matem%C3%A1tica)" title="Matriz (matemática)">Teoria de matrizes</a></li> <li><a href="/wiki/Vetor_(matem%C3%A1tica)" title="Vetor (matemática)">Vetor</a> – <a href="/wiki/Espa%C3%A7o_vetorial" title="Espaço vetorial">Espaço vetorial</a></li> <li><a href="/wiki/Produto_interno" title="Produto interno">Produto interno</a> – <a href="/wiki/Produto_interno" title="Produto interno">Espaço com produto interno</a></li> <li><a href="/wiki/Espa%C3%A7o_de_Hilbert" title="Espaço de Hilbert">Espaço de Hilbert</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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