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mw-first-heading"><span class="mw-page-title-main">Subgrupo comutador</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 16 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-16" 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">16 línguas</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kommutatorgruppe" title="Kommutatorgruppe — alemão" lang="de" hreflang="de" data-title="Kommutatorgruppe" data-language-autonym="Deutsch" data-language-local-name="alemão" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Commutator_subgroup" title="Commutator subgroup — inglês" lang="en" hreflang="en" data-title="Commutator subgroup" data-language-autonym="English" data-language-local-name="inglês" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Subgrupo_conmutador" title="Subgrupo conmutador — espanhol" lang="es" hreflang="es" data-title="Subgrupo conmutador" data-language-autonym="Español" data-language-local-name="espanhol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B2%DB%8C%D8%B1%DA%AF%D8%B1%D9%88%D9%87_%D8%AC%D8%A7%D8%A8%D9%87%E2%80%8C%D8%AC%D8%A7%DA%AF%D8%B1" title="زیرگروه جابه‌جاگر — persa" lang="fa" hreflang="fa" data-title="زیرگروه جابه‌جاگر" 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/Derivaattaryhm%C3%A4" title="Derivaattaryhmä — finlandês" lang="fi" hreflang="fi" data-title="Derivaattaryhmä" 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_d%C3%A9riv%C3%A9" title="Groupe dérivé — francês" lang="fr" hreflang="fr" data-title="Groupe dérivé" data-language-autonym="Français" data-language-local-name="francês" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%AA-%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%95%D7%9E%D7%95%D7%98%D7%98%D7%95%D7%A8%D7%99%D7%9D" title="תת-חבורת הקומוטטורים — hebraico" lang="he" hreflang="he" data-title="תת-חבורת הקומוטטורים" data-language-autonym="עברית" data-language-local-name="hebraico" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Subgrup_komutator" title="Subgrup komutator — indonésio" lang="id" hreflang="id" data-title="Subgrup komutator" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Sottogruppo_derivato" title="Sottogruppo derivato — italiano" lang="it" hreflang="it" data-title="Sottogruppo derivato" 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/%E4%BA%A4%E6%8F%9B%E5%AD%90%E9%83%A8%E5%88%86%E7%BE%A4" title="交換子部分群 — japonês" lang="ja" hreflang="ja" data-title="交換子部分群" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%90%ED%99%98%EC%9E%90_%EB%B6%80%EB%B6%84%EA%B5%B0" title="교환자 부분군 — coreano" lang="ko" hreflang="ko" data-title="교환자 부분군" data-language-autonym="한국어" data-language-local-name="coreano" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Komutant" title="Komutant — polaco" lang="pl" hreflang="pl" data-title="Komutant" data-language-autonym="Polski" data-language-local-name="polaco" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BC%D1%83%D1%82%D0%B0%D0%BD%D1%82" title="Коммутант — russo" lang="ru" hreflang="ru" data-title="Коммутант" data-language-autonym="Русский" data-language-local-name="russo" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D1%83%D1%82%D0%B0%D0%BD%D1%82" title="Комутант — ucraniano" lang="uk" hreflang="uk" data-title="Комутант" data-language-autonym="Українська" data-language-local-name="ucraniano" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_con_giao_ho%C3%A1n_t%E1%BB%AD" title="Nhóm con giao hoán tử — vietnamita" lang="vi" hreflang="vi" data-title="Nhóm con giao hoán tử" 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-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8D%A2%E4%BD%8D%E5%AD%90%E7%BE%A4" title="换位子群 — chinês" lang="zh" hreflang="zh" data-title="换位子群" data-language-autonym="中文" data-language-local-name="chinês" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q522216#sitelinks-wikipedia" title="Editar hiperligações interlínguas" class="wbc-editpage">Editar hiperligações</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaços nominais"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul 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.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=Subgrupo+comutador">Google</a> (<a rel="nofollow" class="external text" href="https://www.google.com/search?hl=pt&tbm=nws&q=Subgrupo+comutador&oq=Subgrupo+comutador">N</a> • <a rel="nofollow" class="external text" href="http://books.google.com/books?&as_brr=0&as_epq=Subgrupo+comutador">L</a> • <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?hl=pt&q=Subgrupo+comutador">A</a>)</span></small></span> <small class="date-container"><i>(<span class="date">Agosto de 2021</span>)</i></small></div></td></tr></tbody></table> <p>Em <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a>, mais especificamente em <a href="/wiki/%C3%81lgebra_abstrata" title="Álgebra abstrata">álgebra abstrata</a>, o <b>subgrupo comutador</b> ou <b>subgrupo derivado</b> de um <a href="/wiki/Grupo_(matem%C3%A1tica)" title="Grupo (matemática)">grupo</a> é o <a href="/wiki/Subgrupo" title="Subgrupo">subgrupo</a> <a href="/wiki/Conjunto_gerador_de_um_grupo" title="Conjunto gerador de um grupo">gerado</a> por todos os <a href="/wiki/Comutador_(matem%C3%A1tica)" title="Comutador (matemática)">comutadores</a> do grupo. Em outras palavras, o comutador de um grupo é o menor <a href="/wiki/Normal" class="mw-disambig" title="Normal">subgrupo normal</a> tal que o quociente é abeliano. </p> <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=Subgrupo_comutador&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=Subgrupo_comutador&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>Um <a href="/wiki/Comutador_(matem%C3%A1tica)" title="Comutador (matemática)">comutador</a> é um elemento da forma <i>g<sup>-1</sup> h<sup>-1</sup> g h</i>, representado por <i>[g, h]</i>. </p><p>O <b>subgrupo comutador</b> é o menor subgrupo que contém todos os comutadores. </p><p>Representa-se o comutador do grupo <i>G</i> por <i>G' </i>. Esta definição pode ser repetida um número finito (ou infinito - usando-se a <a href="/wiki/Recurs%C3%A3o_transfinita" class="mw-redirect" title="Recursão transfinita">recursão transfinita</a>) de vezes, representando-se: </p> <ul><li><i>G<sup>(0)</sup></i> = <i>G</i></li> <li><i>G<sup>(n+1)</sup></i> é o comutador de <i>G<sup>(n)</sup></i></li> <li><i>G<sup>α</sup></i>, para um <a href="/wiki/N%C3%BAmero_ordinal_limite" class="mw-redirect" title="Número ordinal limite">número ordinal limite</a> <i>α</i>, é a interseção dos <i>G<sup>x</sup></i> para todos <i>x < α</i></li></ul> <div class="mw-heading mw-heading2"><h2 id="Propriedades">Propriedades</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Subgrupo_comutador&veaction=edit&section=2" title="Editar secção: Propriedades" class="mw-editsection-visualeditor"><span>editar</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Subgrupo_comutador&action=edit&section=2" title="Editar código-fonte da secção: Propriedades"><span>editar código-fonte</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Se um grupo <i>G</i> é abeliano, então <i>G' = { e }</i></li> <li>O subgrupo comutador é um <a href="/wiki/Subgrupo_normal" title="Subgrupo normal">subgrupo normal</a></li> <li>O quociente <i>G/G' </i> é um grupo abeliano</li> <li>Se <i>N</i> é um subgrupo normal de <i>G</i>, então <i>G/N</i> é abeliano se, e somente se, <i>G' </i> for um subgrupo de <i>N</i></li></ul> <table class="noprint" style="border-top:none; border-bottom:none; background:transparent; color: inherit"><tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Ficheiro:E-to-the-i-pi.svg" class="mw-file-description"><img alt="Ícone de esboço" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/34px-E-to-the-i-pi.svg.png" decoding="async" width="34" height="30" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/51px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/68px-E-to-the-i-pi.svg.png 2x" data-file-width="512" data-file-height="453" /></a></span></td> <td><i>Este artigo sobre <a href="/wiki/Matem%C3%A1tica" title="Matemática">matemática</a> é um <a href="/wiki/Wikip%C3%A9dia:Esbo%C3%A7o" title="Wikipédia:Esboço">esboço</a>. Você pode ajudar a Wikipédia <b><span class="plainlinks"><a class="external text" href="https://pt.wikipedia.org/w/index.php?title=Subgrupo_comutador&action=edit">expandindo-o</a></span></b>.</i><div class="plainlinks hlist navbar mini" style="position:absolute; right:15px; font-size:smaller; display:none;"><ul><li class="nv-ver"><a href="/wiki/Predefini%C3%A7%C3%A3o:Esbo%C3%A7o-matem%C3%A1tica" title="Predefinição:Esboço-matemática"><abbr title="Ver esta predefinição">v</abbr></a></li><li class="nv-discutir"><a href="/wiki/Predefini%C3%A7%C3%A3o_Discuss%C3%A3o:Esbo%C3%A7o-matem%C3%A1tica" title="Predefinição Discussão:Esboço-matemática"><abbr title="Discutir esta predefinição">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:Esbo%C3%A7o-matem%C3%A1tica&action=edit"><abbr title="Editar esta predefinição">e</abbr></a></li></ul></div></td> </tr></tbody></table>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Obtida de "<a dir="ltr" href="https://pt.wikipedia.org/w/index.php?title=Subgrupo_comutador&oldid=61914066">https://pt.wikipedia.org/w/index.php?title=Subgrupo_comutador&oldid=61914066</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Especial:Categorias" title="Especial:Categorias">Categoria</a>: <ul><li><a href="/wiki/Categoria:Teoria_dos_grupos" title="Categoria:Teoria dos grupos">Teoria dos grupos</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Categorias ocultas: <ul><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_fontes_desde_agosto_de_2021" title="Categoria:!Artigos que carecem de fontes desde agosto de 2021">!Artigos que carecem de fontes desde agosto de 2021</a></li><li><a href="/wiki/Categoria:!Artigos_que_carecem_de_fontes_sem_indica%C3%A7%C3%A3o_de_tema" title="Categoria:!Artigos que carecem de fontes sem indicação de tema">!Artigos que carecem de fontes sem indicação de tema</a></li><li><a href="/wiki/Categoria:!Esbo%C3%A7os_sobre_matem%C3%A1tica" title="Categoria:!Esboços sobre matemática">!Esboços sobre matemática</a></li><li><a href="/wiki/Categoria:!Esbo%C3%A7os_maiores_que_1000_bytes" title="Categoria:!Esboços maiores que 1000 bytes">!Esboços maiores que 1000 bytes</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Esta página foi editada pela última vez às 18h43min de 25 de agosto de 2021.</li> <li id="footer-info-copyright">Este texto é disponibilizado nos termos da licença <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.pt">Atribuição-CompartilhaIgual 4.0 Internacional (CC BY-SA 4.0) da Creative Commons</a>; pode estar sujeito a condições adicionais. 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