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href="#Geometri"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Geometri</span> </div> </a> <ul id="toc-Geometri-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representasi_permutasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representasi_permutasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Representasi permutasi</span> </div> </a> <ul id="toc-Representasi_permutasi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aljabar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aljabar"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Aljabar</span> </div> </a> <ul id="toc-Aljabar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teori_grafik" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Teori_grafik"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Teori grafik</span> </div> </a> <ul id="toc-Teori_grafik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Musik" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Musik"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Musik</span> </div> </a> <ul id="toc-Musik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lihat_pula" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lihat_pula"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Lihat pula</span> </div> </a> <ul id="toc-Lihat_pula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referensi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referensi"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Referensi</span> </div> </a> <ul id="toc-Referensi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bacaan_lebih_lanjut" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bacaan_lebih_lanjut"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Bacaan lebih lanjut</span> </div> </a> <ul id="toc-Bacaan_lebih_lanjut-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pranala_luar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Pranala_luar"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Pranala luar</span> </div> </a> <ul id="toc-Pranala_luar-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 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</nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Klein empat grup</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="Pergi ke artikel dalam bahasa lain. Terdapat 28 bahasa" > <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-28" 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">28 bahasa</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D9%83%D9%84%D8%A7%D9%8A%D9%86_%D8%B1%D8%A8%D8%A7%D8%B9%D9%8A%D8%A9_%D8%A7%D9%84%D8%B9%D9%86%D8%A7%D8%B5%D8%B1" title="زمرة كلاين رباعية العناصر – Arab" lang="ar" hreflang="ar" data-title="زمرة كلاين رباعية العناصر" data-language-autonym="العربية" data-language-local-name="Arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_de_Klein" title="Grup de Klein – Katalan" lang="ca" hreflang="ca" data-title="Grup de Klein" data-language-autonym="Català" data-language-local-name="Katalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kleinova_%C4%8Dty%C5%99grupa" title="Kleinova čtyřgrupa – Cheska" lang="cs" hreflang="cs" data-title="Kleinova čtyřgrupa" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Kleins_firegruppe" title="Kleins firegruppe – Dansk" lang="da" hreflang="da" data-title="Kleins firegruppe" data-language-autonym="Dansk" data-language-local-name="Dansk" 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/Kleinsche_Vierergruppe" title="Kleinsche Vierergruppe – Jerman" lang="de" hreflang="de" data-title="Kleinsche Vierergruppe" data-language-autonym="Deutsch" data-language-local-name="Jerman" 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/Klein_four-group" title="Klein four-group – Inggris" lang="en" hreflang="en" data-title="Klein four-group" data-language-autonym="English" data-language-local-name="Inggris" 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/Grupo_de_Klein" title="Grupo de Klein – Spanyol" lang="es" hreflang="es" data-title="Grupo de Klein" data-language-autonym="Español" data-language-local-name="Spanyol" 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/%DA%AF%D8%B1%D9%88%D9%87_%DA%86%D9%87%D8%A7%D8%B1%D8%AA%D8%A7%DB%8C%DB%8C_%DA%A9%D9%84%D8%A7%DB%8C%D9%86" title="گروه چهارتایی کلاین – Persia" lang="fa" hreflang="fa" data-title="گروه چهارتایی کلاین" data-language-autonym="فارسی" data-language-local-name="Persia" 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/Kleinin_neliryhm%C3%A4" title="Kleinin neliryhmä – Suomi" lang="fi" hreflang="fi" data-title="Kleinin neliryhmä" data-language-autonym="Suomi" data-language-local-name="Suomi" 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_de_Klein" title="Groupe de Klein – Prancis" lang="fr" hreflang="fr" data-title="Groupe de Klein" data-language-autonym="Français" data-language-local-name="Prancis" 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%97%D7%91%D7%95%D7%A8%D7%AA_%D7%94%D7%90%D7%A8%D7%91%D7%A2%D7%94_%D7%A9%D7%9C_%D7%A7%D7%9C%D7%99%D7%99%D7%9F" title="חבורת הארבעה של קליין – Ibrani" lang="he" hreflang="he" data-title="חבורת הארבעה של קליין" data-language-autonym="עברית" data-language-local-name="Ibrani" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Klein-csoport" title="Klein-csoport – Hungaria" lang="hu" hreflang="hu" data-title="Klein-csoport" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_de_Klein" title="Gruppo de Klein – Interlingua" lang="ia" hreflang="ia" data-title="Gruppo de Klein" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_di_Klein" title="Gruppo di Klein – Italia" lang="it" hreflang="it" data-title="Gruppo di Klein" data-language-autonym="Italiano" data-language-local-name="Italia" 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/%E3%82%AF%E3%83%A9%E3%82%A4%E3%83%B3%E3%81%AE%E5%9B%9B%E5%85%83%E7%BE%A4" title="クラインの四元群 – Jepang" lang="ja" hreflang="ja" data-title="クラインの四元群" data-language-autonym="日本語" data-language-local-name="Jepang" 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/%ED%81%B4%EB%9D%BC%EC%9D%B8_4%EC%9B%90%EA%B5%B0" title="클라인 4원군 – Korea" lang="ko" hreflang="ko" data-title="클라인 4원군" data-language-autonym="한국어" data-language-local-name="Korea" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B2%E0%B5%88%E0%B5%BB_%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="ക്ലൈൻ ഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ക്ലൈൻ ഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Viergroep_van_Klein" title="Viergroep van Klein – Belanda" lang="nl" hreflang="nl" data-title="Viergroep van Klein" data-language-autonym="Nederlands" data-language-local-name="Belanda" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_czw%C3%B3rkowa_Kleina" title="Grupa czwórkowa Kleina – Polski" lang="pl" hreflang="pl" data-title="Grupa czwórkowa Kleina" data-language-autonym="Polski" data-language-local-name="Polski" 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_%C3%ABd_Klein" title="Strop ëd Klein – Piedmontese" lang="pms" hreflang="pms" data-title="Strop ëd Klein" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Klein_4" title="Klein 4 – Portugis" lang="pt" hreflang="pt" data-title="Klein 4" data-language-autonym="Português" data-language-local-name="Portugis" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Grupul_lui_Klein" title="Grupul lui Klein – Rumania" lang="ro" hreflang="ro" data-title="Grupul lui Klein" data-language-autonym="Română" data-language-local-name="Rumania" 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%A7%D0%B5%D1%82%D0%B2%D0%B5%D1%80%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0_%D0%9A%D0%BB%D0%B5%D0%B9%D0%BD%D0%B0" title="Четверная группа Клейна – Rusia" lang="ru" hreflang="ru" data-title="Четверная группа Клейна" data-language-autonym="Русский" data-language-local-name="Rusia" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kleins_fyrgrupp" title="Kleins fyrgrupp – Swedia" lang="sv" hreflang="sv" data-title="Kleins fyrgrupp" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BF%E0%AE%B3%E0%AF%88%E0%AE%A9%E0%AF%8D_%E0%AE%A8%E0%AE%BE%E0%AE%A9%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="கிளைன் நான்குறுப்புக்குலம் – Tamil" lang="ta" hreflang="ta" data-title="கிளைன் நான்குறுப்புக்குலம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/4-%D0%B3%D1%80%D1%83%D0%BF%D0%B0_%D0%9A%D0%BB%D1%8F%D0%B9%D0%BD%D0%B0" title="4-група Кляйна – Ukraina" lang="uk" hreflang="uk" data-title="4-група Кляйна" data-language-autonym="Українська" data-language-local-name="Ukraina" 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_b%E1%BB%91n_Klein" title="Nhóm bốn Klein – Vietnam" lang="vi" hreflang="vi" data-title="Nhóm bốn Klein" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnam" 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/%E5%85%8B%E8%8E%B1%E5%9B%A0%E5%9B%9B%E5%85%83%E7%BE%A4" title="克莱因四元群 – Tionghoa" lang="zh" hreflang="zh" data-title="克莱因四元群" data-language-autonym="中文" data-language-local-name="Tionghoa" 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/Q550593#sitelinks-wikipedia" title="Sunting pranala interwiki" class="wbc-editpage">Sunting pranala</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="Ruang nama"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Klein_empat_grup" title="Lihat halaman isi [c]" accesskey="c"><span>Halaman</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Pembicaraan:Klein_empat_grup" rel="discussion" title="Pembicaraan halaman isi [t]" accesskey="t"><span>Pembicaraan</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ubah varian bahasa" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Bahasa Indonesia</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Tampilan"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Klein_empat_grup"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&action=history" title="Revisi sebelumnya dari halaman ini. [h]" accesskey="h"><span>Lihat riwayat</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Perkakas" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Perkakas</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header 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vector-more-collapsible-item mw-list-item"><a href="/wiki/Klein_empat_grup"><span>Baca</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&veaction=edit" title="Sunting halaman ini [v]" accesskey="v"><span>Sunting</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&action=edit" title="Sunting kode sumber halaman ini [e]" accesskey="e"><span>Sunting sumber</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&action=history"><span>Lihat riwayat</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Umum </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Istimewa:Pranala_balik/Klein_empat_grup" title="Daftar semua halaman wiki yang memiliki pranala ke halaman ini [j]" accesskey="j"><span>Pranala balik</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Istimewa:Perubahan_terkait/Klein_empat_grup" rel="nofollow" title="Perubahan terbaru halaman-halaman yang memiliki pranala ke halaman ini [k]" accesskey="k"><span>Perubahan terkait</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Istimewa:Halaman_istimewa" title="Daftar semua halaman istimewa [q]" accesskey="q"><span>Halaman istimewa</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&oldid=24009583" title="Pranala permanen untuk revisi halaman ini"><span>Pranala permanen</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&action=info" title="Informasi lanjut tentang halaman ini"><span>Informasi halaman</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Kutip&page=Klein_empat_grup&id=24009583&wpFormIdentifier=titleform" title="Informasi tentang bagaimana mengutip halaman ini"><span>Kutip halaman ini</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Istimewa:UrlShortener&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FKlein_empat_grup"><span>Lihat URL pendek</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Istimewa:QrCode&url=https%3A%2F%2Fid.wikipedia.org%2Fwiki%2FKlein_empat_grup"><span>Unduh kode QR</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Cetak/ekspor </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Istimewa:Buku&bookcmd=book_creator&referer=Klein+empat+grup"><span>Buat buku</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Istimewa:DownloadAsPdf&page=Klein_empat_grup&action=show-download-screen"><span>Unduh versi PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Klein_empat_grup&printable=yes" title="Versi cetak halaman ini [p]" accesskey="p"><span>Versi cetak</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Dalam proyek lain </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Klein_four-group" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q550593" title="Pranala untuk menghubungkan butir pada ruang penyimpanan data [g]" accesskey="g"><span>Butir di Wikidata</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Peralatan halaman"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Tampilan"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Tampilan</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sembunyikan</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Dari Wikipedia bahasa Indonesia, ensiklopedia bebas</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="id" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r18844875">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable">Halaman ini berisi artikel tentang konsep matematika. Untuk empat orang grup Perlawanan anti-Nazi, lihat <a href="/w/index.php?title=Vierergruppe_(Perlawanan_Jerman)&action=edit&redlink=1" class="new" title="Vierergruppe (Perlawanan Jerman) (halaman belum tersedia)">Vierergruppe (Perlawanan Jerman)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Artikel ini bukan mengenai <a href="/w/index.php?title=Grup_Kleinian&action=edit&redlink=1" class="new" title="Grup Kleinian (halaman belum tersedia)">Grup Kleinian</a>, sub-grup terpisah dari PSL(2, C).</div> <style data-mw-deduplicate="TemplateStyles:r26333518">.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Struktur_aljabar" title="Struktur aljabar">Struktur aljabar</a> → <b>Teori grup</b></span><br /><a href="/wiki/Teori_grup" title="Teori grup">Teori grup</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/Berkas:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;">Gagasan dasar</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgrup" title="Subgrup">Subgrup</a></li> <li><a href="/wiki/Subgrup_normal" title="Subgrup normal">Subgrup normal</a></li></ul> <ul><li><a href="/wiki/Grup_hasil_bagi" title="Grup hasil bagi">Grup hasil bagi</a></li> <li><a href="/w/index.php?title=Grup_darab_langsung&action=edit&redlink=1" class="new" title="Grup darab langsung (halaman belum tersedia)">darab langsung</a></li> <li><a href="/w/index.php?title=Grup_semi-darab_langsung&action=edit&redlink=1" class="new" title="Grup semi-darab langsung (halaman belum tersedia)">semi-darab langsung</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Homomorfisme_grup" class="mw-redirect" title="Homomorfisme grup">Homomorfisme grup</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(aljabar)#Homomorfisme_grup" title="Kernel (aljabar)">kernel</a></li> <li><a href="/wiki/Bayangan_(matematika)" title="Bayangan (matematika)">bayangan</a></li> <li><a href="/w/index.php?title=Jumlah_grup_langsung&action=edit&redlink=1" class="new" title="Jumlah grup langsung (halaman belum tersedia)">jumlah langsung</a></li></ul> <ul><li><a href="/w/index.php?title=Darab_karangan_bunga&action=edit&redlink=1" class="new" title="Darab karangan bunga (halaman belum tersedia)">karangan bunga</a></li> <li><a href="/wiki/Grup_sederhana" title="Grup sederhana">sederhana</a></li> <li><a href="/wiki/Grup_hingga" title="Grup hingga">hingga</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_takhingga&action=edit&redlink=1" class="new" title="Grup takhingga (halaman belum tersedia)">takhingga</a></li> <li><a href="/w/index.php?title=Grup_kontinu&action=edit&redlink=1" class="new" title="Grup kontinu (halaman belum tersedia)">kontinu</a></li> <li><a href="/w/index.php?title=Grup_multiplikatif&action=edit&redlink=1" class="new" title="Grup multiplikatif (halaman belum tersedia)">multiplikatif</a></li></ul> <ul><li><a href="/wiki/Grup_aditif" title="Grup aditif">aditif</a></li> <li><a href="/wiki/Grup_siklik" title="Grup siklik">siklik</a></li> <li><a href="/w/index.php?title=Grup_Abel&action=edit&redlink=1" class="new" title="Grup Abel (halaman belum tersedia)">Abel</a></li> <li><a href="/wiki/Grup_dihedral" title="Grup dihedral">dihedral</a></li></ul> <ul><li><a href="/wiki/Grup_nilpoten" title="Grup nilpoten">nilpoten</a></li> <li><a href="/w/index.php?title=Grup_terselesaikan&action=edit&redlink=1" class="new" title="Grup terselesaikan (halaman belum tersedia)">terselesaikan</a></li> <li><a href="/w/index.php?title=Aksi_grup&action=edit&redlink=1" class="new" title="Aksi grup (halaman belum tersedia)">aksi</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/w/index.php?title=Glosarium_teori_grup&action=edit&redlink=1" class="new" title="Glosarium teori grup (halaman belum tersedia)">Glosarium teori grup</a></li> <li><a href="/wiki/Daftar_topik_teori_grup" title="Daftar topik teori grup">Daftar topik teori grup</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/wiki/Grup_hingga" title="Grup hingga">Grup hingga</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r26333518"><table class="sidebar" style="border-collapse: collapse; border-spacing: 0px; border:none; width:100%; margin:0px; font-size: 100%; clear:none; float:none;"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Klasifikasi_grup_sederhana_hingga" title="Klasifikasi grup sederhana hingga">Klasifikasi grup sederhana hingga</a></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Grup_siklik" title="Grup siklik">siklik</a></li> <li><a href="/w/index.php?title=Grup_bergantian&action=edit&redlink=1" class="new" title="Grup bergantian (halaman belum tersedia)">bergantian</a></li> <li><a href="/w/index.php?title=Grup_tipe_Lie&action=edit&redlink=1" class="new" title="Grup tipe Lie (halaman belum tersedia)">tipe Lie</a></li> <li><a href="/w/index.php?title=Grup_sporadik&action=edit&redlink=1" class="new" title="Grup sporadik (halaman belum tersedia)">sporadik</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/w/index.php?title=Teorema_Cauchy_(teori_grup)&action=edit&redlink=1" class="new" title="Teorema Cauchy (teori grup) (halaman belum tersedia)">Teorema Cauchy</a></li> <li><a href="/wiki/Teorema_Lagrange_(teori_grup)" title="Teorema Lagrange (teori grup)">Teorema Lagrange</a></li></ul> <ul><li><a href="/wiki/Teorema_Sylow" title="Teorema Sylow">Teorema Sylow</a></li> <li><a href="/w/index.php?title=Subgrup_Hall&action=edit&redlink=1" class="new" title="Subgrup Hall (halaman belum tersedia)">Teorema Hall</a></li></ul> <ul><li><a href="/wiki/Grup-p" title="Grup-p">grup-p</a></li> <li><a href="/w/index.php?title=Grup_Abel_elementer&action=edit&redlink=1" class="new" title="Grup Abel elementer (halaman belum tersedia)">Grup Abel elementer</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_Frobenius&action=edit&redlink=1" class="new" title="Grup Frobenius (halaman belum tersedia)">Grup Frobenius</a></li></ul> <ul><li><a href="/w/index.php?title=Pengganda_Schur&action=edit&redlink=1" class="new" title="Pengganda Schur (halaman belum tersedia)">Pengganda Schur</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Grup_simetrik" title="Grup simetrik">Grup simetrik</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 \mathrm {S} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {S} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/412f98267cda84a9c8abfee60f7184af3cb1aeb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.511ex; height:2.509ex;" alt="{\displaystyle \mathrm {S} _{n}}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_Klein&action=edit&redlink=1" class="new" title="Grup Klein (halaman belum tersedia)">Grup Klein</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 \mathrm {V} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {V} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/664209da7650f00b3507efe25f89aeff9783146c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {V} }"></span></li> <li><a href="/wiki/Grup_dihedral" title="Grup dihedral">Grup dihedral</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 \mathrm {D} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {D} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ded8c7d71e610ba30a0856fa881290ae80b7282b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.994ex; height:2.509ex;" alt="{\displaystyle \mathrm {D} _{n}}"></span></li> <li><a href="/wiki/Grup_kuaternion" title="Grup kuaternion">Grup kuaternion</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 \mathrm {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31ea6b6e5d15ac13060c9724fdbf3aa79b353f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathrm {Q} }"></span></li> <li><a href="/wiki/Grup_disiklik" title="Grup disiklik">Grup disiklik</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 \mathrm {Dic} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Dic} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/911cef2c1b11010e151d3737a8f3b8588f3b00e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.673ex; height:2.509ex;" alt="{\displaystyle \mathrm {Dic} _{n}}"></span></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><style data-mw-deduplicate="TemplateStyles:r23782733">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><div class="hlist"><ul><li><a href="/w/index.php?title=Grup_diskret&action=edit&redlink=1" class="new" title="Grup diskret (halaman belum tersedia)">Grup diskret</a></li><li><a href="/w/index.php?title=Kekisi_(subgrup_diskret)&action=edit&redlink=1" class="new" title="Kekisi (subgrup diskret) (halaman belum tersedia)">Kekisi</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Bilangan_bulat" title="Bilangan bulat">Bilangan bulat</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Grup_bebas" title="Grup bebas">Grup bebas</a></li></ul> <div style="padding:0.2em 0.4em; line-height:1.2em;"><a href="/w/index.php?title=Grup_modular&action=edit&redlink=1" class="new" title="Grup modular (halaman belum tersedia)">Grup modular</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><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 \mathrm {PSL} (2,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f4d0e8493b732b05e29613a714405de8b25356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.884ex; height:2.843ex;" alt="{\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}"></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 \mathrm {SL} (2,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (2,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe790f9dda6d5d14bf20e9f5f92d9b0ff83e696" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.301ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (2,\mathbb {Z} )}"></span></li></ul></div></div> <ul><li><a href="/w/index.php?title=Grup_aritmetika&action=edit&redlink=1" class="new" title="Grup aritmetika (halaman belum tersedia)">Grup aritmetika</a></li> <li><a href="/w/index.php?title=Kekisi_(grup)&action=edit&redlink=1" class="new" title="Kekisi (grup) (halaman belum tersedia)">Kekisi</a></li> <li><a href="/w/index.php?title=Grup_hiperbolik&action=edit&redlink=1" class="new" title="Grup hiperbolik (halaman belum tersedia)">Grup hiperbolik</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/w/index.php?title=Grup_topologis&action=edit&redlink=1" class="new" title="Grup topologis (halaman belum tersedia)">Topologis</a> dan <a href="/wiki/Grup_Lie" title="Grup Lie">Grup Lie</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/w/index.php?title=Solenoid_(matematika)&action=edit&redlink=1" class="new" title="Solenoid (matematika) (halaman belum tersedia)">Solenoid</a></li> <li><a href="/w/index.php?title=Grup_lingkaran&action=edit&redlink=1" class="new" title="Grup lingkaran (halaman belum tersedia)">Lingkaran</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_linear_umum&action=edit&redlink=1" class="new" title="Grup linear umum (halaman belum tersedia)">Linear umum</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 \mathrm {GL} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {GL} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba99fb253ee3b9082e5d718da746260073e6c7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.481ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_linear_khusus&action=edit&redlink=1" class="new" title="Grup linear khusus (halaman belum tersedia)">Linear khusus</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 \mathrm {SL} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SL} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cd269a9439a1075bb460bf6ae7b1407086c35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.949ex; height:2.843ex;" alt="{\displaystyle \mathrm {SL} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_ortogonal&action=edit&redlink=1" class="new" title="Grup ortogonal (halaman belum tersedia)">Ortogonal</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 \mathrm {O} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {O} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1471779b64c8868583dcd50e3c6381293f0dd67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.012ex; height:2.843ex;" alt="{\displaystyle \mathrm {O} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_Euklides&action=edit&redlink=1" class="new" title="Grup Euklides (halaman belum tersedia)">Euklides</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 \mathrm {E} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {E} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f699fb0fd3801a34a5016afa897a24953fa9aab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.787ex; height:2.843ex;" alt="{\displaystyle \mathrm {E} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_ortogonal_khusus&action=edit&redlink=1" class="new" title="Grup ortogonal khusus (halaman belum tersedia)">Ortogonal khusus</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 \mathrm {SO} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">O</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SO} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa71842f19b6810b4bfa9eb282e92fbf285094e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \mathrm {SO} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_uner&action=edit&redlink=1" class="new" title="Grup uner (halaman belum tersedia)">Uner</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 \mathrm {U} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {U} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32fa84df5de5dfa91b6bdc88fb03fc8792c9f81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \mathrm {U} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_uniter_khusus&action=edit&redlink=1" class="new" title="Grup uniter khusus (halaman belum tersedia)">Uniter khusus</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 \mathrm {SU} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8a205091aabd5690efdfeb7354a55844f2eb31b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=Grup_simplektik&action=edit&redlink=1" class="new" title="Grup simplektik (halaman belum tersedia)">Simplektik</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 \mathrm {Sp} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Sp} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00b37fb88bcb5053f70e6386aa2119328bd1171" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle \mathrm {Sp} (n)}"></span></li></ul> <ul><li><a href="/w/index.php?title=G2_(matematika)&action=edit&redlink=1" class="new" title="G2 (matematika) (halaman belum tersedia)">G<sub>2</sub></a></li> <li><a href="/w/index.php?title=F4_(matematika)&action=edit&redlink=1" class="new" title="F4 (matematika) (halaman belum tersedia)">F<sub>4</sub></a></li> <li><a href="/w/index.php?title=E6_(matematika)&action=edit&redlink=1" class="new" title="E6 (matematika) (halaman belum tersedia)">E<sub>6</sub></a></li> <li><a href="/w/index.php?title=E7_(matematika)&action=edit&redlink=1" class="new" title="E7 (matematika) (halaman belum tersedia)">E<sub>7</sub></a></li> <li><a href="/w/index.php?title=E8_(matematika)&action=edit&redlink=1" class="new" title="E8 (matematika) (halaman belum tersedia)">E<sub>8</sub></a></li></ul> <ul><li><a href="/w/index.php?title=Grup_Loretnz&action=edit&redlink=1" class="new" title="Grup Loretnz (halaman belum tersedia)">Lorentz</a></li> <li><a href="/w/index.php?title=Grup_Poincar%C3%A9&action=edit&redlink=1" class="new" title="Grup Poincaré (halaman belum tersedia)">Poincaré</a></li> <li><a href="/w/index.php?title=Group_konformal&action=edit&redlink=1" class="new" title="Group konformal (halaman belum tersedia)">konformal</a></li></ul> <ul><li><a href="/w/index.php?title=Difeomorfisme&action=edit&redlink=1" class="new" title="Difeomorfisme (halaman belum tersedia)">Difeomorfisme</a></li> <li><a href="/w/index.php?title=Grup_gelung&action=edit&redlink=1" class="new" title="Grup gelung (halaman belum tersedia)">Gelung</a></li></ul> <div style="padding:0.2em 0.4em; line-height:1.2em;"><a href="/w/index.php?title=Grup_Lie_berdimensi_takhingga&action=edit&redlink=1" class="new" title="Grup Lie berdimensi takhingga (halaman belum tersedia)">Grup Lie berdimensi takhingga</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r23782733"><div class="hlist"><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 O(\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b333ad74141a87280c1fbe4ae31d7bc0dcc572a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.906ex; height:2.843ex;" alt="{\displaystyle O(\infty )}"></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 \mathrm {SU} (\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">U</mi> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {SU} (\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c73a9f1ca8515559b3a55fd76827f11457af591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.169ex; height:2.843ex;" alt="{\displaystyle \mathrm {SU} (\infty )}"></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 \mathrm {Sp} (\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">p</mi> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Sp} (\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92382a0fb0bdd299850c5505365353df0b04a921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle \mathrm {Sp} (\infty )}"></span></li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;"><a href="/w/index.php?title=Grup_aljabar&action=edit&redlink=1" class="new" title="Grup aljabar (halaman belum tersedia)">Grup aljabar</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/w/index.php?title=Grup_aljabar_linear&action=edit&redlink=1" class="new" title="Grup aljabar linear (halaman belum tersedia)">Grup aljabar linear</a></li></ul> <ul><li><a href="/w/index.php?title=Grup_reduktif&action=edit&redlink=1" class="new" title="Grup reduktif (halaman belum tersedia)">Grup reduktif</a></li></ul> <ul><li><a href="/w/index.php?title=Varietas_Abel&action=edit&redlink=1" class="new" title="Varietas Abel (halaman belum tersedia)">Varietas Abel</a></li></ul> <ul><li><a href="/wiki/Kurva_eliptik" title="Kurva eliptik">Kurva eliptik</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r18590415">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-lihat"><a href="/wiki/Templat:Teori_grup_sidebar" title="Templat:Teori grup sidebar"><abbr title="Lihat templat ini">l</abbr></a></li><li class="nv-bicara"><a href="/wiki/Pembicaraan_Templat:Teori_grup_sidebar" title="Pembicaraan Templat:Teori grup sidebar"><abbr title="Diskusikan templat ini">b</abbr></a></li><li class="nv-sunting"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Templat:Teori_grup_sidebar&action=edit"><abbr title="Sunting templat ini">s</abbr></a></li></ul></div></td></tr></tbody></table> <p>Dalam <a href="/wiki/Matematika" title="Matematika">matematika</a>, <b>Klein empat grup</b> adalah <a href="/wiki/Grup_(matematika)" title="Grup (matematika)"> grup</a> dengan empat elemen, di mana setiap elemen adalah <a href="/w/index.php?title=Involution_(matematika)&action=edit&redlink=1" class="new" title="Involution (matematika) (halaman belum tersedia)"> self-inverse</a> (menyusunnya dengan sendirinya menghasilkan identitas) dan di mana menyusun dua dari tiga elemen non-identitas menghasilkan yang ketiga. Ini dapat dideskripsikan sebagai <a href="/w/index.php?title=Grup_simetri&action=edit&redlink=1" class="new" title="Grup simetri (halaman belum tersedia)">grup simetri</a> dari non-persegi <a href="/wiki/Persegi_panjang" title="Persegi panjang">persegi panjang</a> (dengan tiga elemen non-identitas menjadi refleksi horizontal dan vertikal dan rotasi 180 derajat), sebagai grup operasi <a href="/w/index.php?title=Operasi_Bitwise&action=edit&redlink=1" class="new" title="Operasi Bitwise (halaman belum tersedia)"> bitwise</a> <a href="/w/index.php?title=Eksklusif_atau&action=edit&redlink=1" class="new" title="Eksklusif atau (halaman belum tersedia)">eksklusif atau</a> pada nilai biner dua bit, atau lebih <a href="/wiki/Aljabar_abstrak" title="Aljabar abstrak"> abstrak</a> sebagai <span class="nowrap">Z<sub>2</sub> × Z<sub>2</sub></span>, <a href="/w/index.php?title=Produk_langsung_dari_grup&action=edit&redlink=1" class="new" title="Produk langsung dari grup (halaman belum tersedia)"> produk langsung</a> dari dua salinan dari <a href="/wiki/Grup_siklik" title="Grup siklik">grup siklik</a> dari <a href="/w/index.php?title=Urutan_(teori_grup)&action=edit&redlink=1" class="new" title="Urutan (teori grup) (halaman belum tersedia)"> pesanan</a> 2. Nama <i><b>Vierergruppe</b></i> (yang berarti empat grup) oleh <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> pada tahun 1884.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Ini juga disebut <b>Grup Klein</b>, dan sering dilambangkan dengan huruf V atau sebagai K<sub>4</sub>. </p><p>Grup empat Klein, dengan empat elemen, adalah grup terkecil yang bukan merupakan <a href="/wiki/Grup_siklik" title="Grup siklik">grup siklik</a>. Hanya ada satu grup lain dari orde empat, hingga <a href="/w/index.php?title=Grup_isomorfisme&action=edit&redlink=1" class="new" title="Grup isomorfisme (halaman belum tersedia)"> isomorfisme</a>, grup siklik urutan 4. Keduanya adalah <a href="/wiki/Grup_abelian" class="mw-redirect" title="Grup abelian">grup abelian</a>. Golongan non-abelian terkecil adalah <a href="/w/index.php?title=Golongan_simetris_derajat_3&action=edit&redlink=1" class="new" title="Golongan simetris derajat 3 (halaman belum tersedia)">golongan simetris derajat 3</a>, yang berurutan 6. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Presentasi">Presentasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=1" title="Sunting bagian: Presentasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=1" title="Sunting kode sumber bagian: Presentasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Tabel_Cayley&action=edit&redlink=1" class="new" title="Tabel Cayley (halaman belum tersedia)">Tabel Cayley</a> dari grup Klein diberikan oleh: </p> <table class="wikitable" width="120"> <tbody><tr> <th>* </th> <th>e </th> <th><i>a</i> </th> <th><i>b</i> </th> <th><i>c</i> </th></tr> <tr align="center"> <th>e </th> <td>e</td> <td><i>a</i></td> <td><i>b</i></td> <td><i>c</i> </td></tr> <tr align="center"> <th><i>a</i> </th> <td><i>a</i></td> <td>e</td> <td><i>c</i></td> <td><i>b</i> </td></tr> <tr align="center"> <th><i>b</i> </th> <td><i>b</i></td> <td><i>c</i></td> <td>e</td> <td><i>a</i> </td></tr> <tr align="center"> <th><i>c</i> </th> <td><i>c</i></td> <td><i>b</i></td> <td><i>a</i></td> <td>e </td></tr></tbody></table> <p>Empat kelompok Klein juga ditentukan oleh <a href="/wiki/Presentasi_grup" title="Presentasi grup">presentasi grup</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {V} =\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=e\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> </mrow> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∣</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>e</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {V} =\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=e\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60024dba0dfdb1fea565b682a9e721b86ce6ade5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.302ex; height:3.176ex;" alt="{\displaystyle \mathrm {V} =\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=e\rangle .}"></span></dd></dl> <p>Semua elemen non - <a href="/wiki/Elemen_identitas" class="mw-redirect" title="Elemen identitas"> identitas</a> dari grup Klein memiliki urutan 2, sehingga dua elemen non-identitas mana pun dapat berfungsi sebagai generator dalam presentasi di atas. Empat kelompok Klein adalah non-<a href="/wiki/Grup_siklik" title="Grup siklik">grup siklik</a> terkecil. Namun ini adalah <a href="/wiki/Kelompok_abelian" class="mw-redirect" title="Kelompok abelian">kelompok abelian</a>, dan isomorfik ke <a href="/w/index.php?title=Kelompok_dihedral&action=edit&redlink=1" class="new" title="Kelompok dihedral (halaman belum tersedia)">kelompok dihedral</a> urutan (kardinalitas) 4, yaitu D<sub>4</sub> (atau D<sub>2</sub>, menggunakan konvensi geometris); selain grup urutan 2, itu adalah satu-satunya grup dihedral yang abelian. </p><p>Grup empat Klein juga isomorfik terhadap <a href="/wiki/Jumlah_langsung" title="Jumlah langsung">jumlah langsung</a> <span class="nowrap">Z<sub>2</sub> ⊕ Z<sub>2</sub></span>, sehingga bisa direpresentasikan sebagai pasangan <span class="nowrap">{(0,0), (0,1), (1,0), (1,1)} </span> di bawah penambahan berdasarkan komponen <a href="/w/index.php?title=Aritmrtika_modular&action=edit&redlink=1" class="new" title="Aritmrtika modular (halaman belum tersedia)"> modulo 2</a> (atau yang setara dengan <a href="/w/index.php?title=Bit_array&action=edit&redlink=1" class="new" title="Bit array (halaman belum tersedia)"> bit strings</a> <span class="nowrap">{00, 01, 10, 11} </span>di bawah <a href="/w/index.php?title=Bitwise_XOR&action=edit&redlink=1" class="new" title="Bitwise XOR (halaman belum tersedia)">bitwise XOR</a>); dengan (0,0) menjadi elemen identitas kelompok. Jadi, empat kelompok Klein adalah contoh dari <a href="/w/index.php?title=Grup_abelian_dasar&action=edit&redlink=1" class="new" title="Grup abelian dasar (halaman belum tersedia)"> kelompok abelian dasar 2</a>, yang juga disebut <a href="/w/index.php?title=Grup_Boolean&action=edit&redlink=1" class="new" title="Grup Boolean (halaman belum tersedia)">grup Boolean</a>. Kelompok empat Klein dengan demikian juga kelompok yang dihasilkan oleh <a href="/wiki/Perbedaan_simetris" class="mw-redirect" title="Perbedaan simetris">perbedaan simetris</a> sebagai operasi biner pada <a href="/wiki/Himpunan_bagian" title="Himpunan bagian">himpunan bagian</a> dari <a href="/w/index.php?title=Himpunan_kekuatan&action=edit&redlink=1" class="new" title="Himpunan kekuatan (halaman belum tersedia)">himpunan kekuatan</a> dari suatu himpunan dengan dua elemen, yaitu di atas <a href="/w/index.php?title=Bidang_himpunan&action=edit&redlink=1" class="new" title="Bidang himpunan (halaman belum tersedia)">bidang himpunan</a> dengan empat elemen, misalnya<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 \{\emptyset ,\{\alpha \},\{\beta \},\{\alpha ,\beta \}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi mathvariant="normal">∅</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>β</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\emptyset ,\{\alpha \},\{\beta \},\{\alpha ,\beta \}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/945b17700beccebe115759ec23715839ddf2af28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.237ex; height:2.843ex;" alt="{\displaystyle \{\emptyset ,\{\alpha \},\{\beta \},\{\alpha ,\beta \}\}}"></span>; <a href="/wiki/Himpunan_kosong" title="Himpunan kosong">himpunan kosong</a> adalah elemen identitas grup dalam kasus ini. </p><p>Konstruksi numerik lain dari grup empat Klein adalah himpunan <span class="nowrap">{ 1, 3, 5, 7 },</span> dengan operasi menjadi <a href="/w/index.php?title=Grup_perkalian_bilangan_bulat_modulo_n&action=edit&redlink=1" class="new" title="Grup perkalian bilangan bulat modulo n (halaman belum tersedia)"> perkalian modulo 8</a>. Di sini <i> a </i> adalah 3, <i> b </i> adalah 5, dan <span class="nowrap"><i>c</i> = <i>ab</i></span> adalah <span class="nowrap">3 × 5 = 15 ≡ 7 (mod 8)</span>. </p><p>Empat grup Klein memiliki representasi sebagai matriks nyata 2x2 dengan operasi perkalian matriks: </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 e={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,\,a={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,\,b={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}\,\,c={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>a</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>b</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,\,a={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,\,b={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}\,\,c={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6785cc9cf51cc73370f04de778245f6fe31b056e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.547ex; height:6.176ex;" alt="{\displaystyle e={\begin{pmatrix}1&0\\0&1\end{pmatrix}}\,\,a={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,\,b={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}\,\,c={\begin{pmatrix}-1&0\\0&-1\end{pmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Geometri">Geometri</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=2" title="Sunting bagian: Geometri" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=2" title="Sunting kode sumber bagian: Geometri"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Secara geometris, dalam dua dimensi, empat grup Klein adalah <a href="/w/index.php?title=Grup_simetri&action=edit&redlink=1" class="new" title="Grup simetri (halaman belum tersedia)">grup simetri</a> dari <a href="/wiki/Belah_ketupat" title="Belah ketupat">belah ketupat</a> dan <a href="/wiki/Persegi_panjang" title="Persegi panjang">persegi panjang</a> yang bukan <a href="/w/index.php?title=Persegi_(geometri)&action=edit&redlink=1" class="new" title="Persegi (geometri) (halaman belum tersedia)"> persegi</a>, Empat elemen tersebut adalah identitas, refleksi vertikal, refleksi horizontal, dan rotasi 180 derajat. </p><p>Dalam tiga dimensi, ada tiga kelompok simetri berbeda yang secara aljabar merupakan empat grup Klein V: </p> <ul><li>satu dengan tiga sumbu rotasi 2 kali lipat tegak lurus: D<sub> 2 </sub></li> <li>satu dengan sumbu rotasi 2 kali lipat, dan bidang refleksi tegak lurus: <span class="nowrap">C<sub>2<i>h</i></sub> = D<sub>1<i>d</i></sub></span></li> <li>satu dengan sumbu rotasi 2 kali lipat dalam bidang refleksi (dan karenanya juga dalam bidang refleksi tegak lurus): <span class="nowrap">C<sub>2<i>v</i></sub> = D<sub>1<i>h</i></sub></span>.</li></ul> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Representasi_permutasi">Representasi permutasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=3" title="Sunting bagian: Representasi permutasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=3" title="Sunting kode sumber bagian: Representasi permutasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,7,16,23).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/220px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/330px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg/440px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C7%2C16%2C23%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Identitas dan ganda - <a href="/w/index.php?title=Transposisi_(matematika)&action=edit&redlink=1" class="new" title="Transposisi (matematika) (halaman belum tersedia)"> transposisi</a> dari empat objek membentuk V</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Berkas:Klein_four-group;_Cayley_table;_subgroup_of_S4_(elements_0,1,6,7).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/220px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/330px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/97/Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg/440px-Klein_four-group%3B_Cayley_table%3B_subgroup_of_S4_%28elements_0%2C1%2C6%2C7%29.svg.png 2x" data-file-width="177" data-file-height="177" /></a><figcaption>Permutasi lain dari empat objek, membentuk V juga<br /><br />Lihat: <a href="/wiki/Subgrup#4_elemen" title="Subgrup"> 4 himpunan bagian elemen dari S<sub>4</sub></a></figcaption></figure> <p>Tiga elemen orde dua dalam grup empat Klein dapat dipertukarkan: <a href="/wiki/Grup_automorfisme" title="Grup automorfisme">grup automorfisme</a> dari V adalah grup permutasi dari ketiga elemen ini. </p><p>Permutasi empat kelompok Klein dari elemennya sendiri dapat dianggap secara abstrak sebagai <a href="/w/index.php?title=Representasi_permutasi&action=edit&redlink=1" class="new" title="Representasi permutasi (halaman belum tersedia)">representasi permutasi</a> pada empat poin: </p> <dl><dd>V = { (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) }</dd></dl> <p>Dalam representasi ini, V adalah <a href="/wiki/Subgrup_normal" title="Subgrup normal">subgrup normal</a> dari <a href="/w/index.php?title=Grup_bergantian&action=edit&redlink=1" class="new" title="Grup bergantian (halaman belum tersedia)">grup bergantian</a> A<sub>4</sub> (dan juga <a href="/wiki/Grup_simetris" class="mw-redirect" title="Grup simetris">grup simetris</a> S<sub>4</sub>) pada empat huruf. Nyatanya, ini adalah <a href="/wiki/Kernel_(aljabar)#Homomorfisme_grup" title="Kernel (aljabar)"> kernel</a> dari sebuah surjektif <a href="/w/index.php?title=Group_homomorphism&action=edit&redlink=1" class="new" title="Group homomorphism (halaman belum tersedia)">group homomorphism</a> dari S<sub>4</sub> ke S<sub>3</sub>. </p><p>Representasi lain di dalam S<sub>4</sub> adalah: </p><p>{ (), (1,2), (3,4), (1,2)(3,4)} </p><p>{ (), (1,3), (2,4), (1,3)(2,4)} </p><p>{ (), (1,4), (2,3), (1,4)(2,3)} </p><p>Mereka bukan subgrup normal dari S<sub>4.</sub> </p> <div style="clear:both;"></div> <div class="mw-heading mw-heading2"><h2 id="Aljabar">Aljabar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=4" title="Sunting bagian: Aljabar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=4" title="Sunting kode sumber bagian: Aljabar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Menurut <a href="/wiki/Teori_Galois" title="Teori Galois">teori Galois</a>, keberadaan empat kelompok Klein (dan khususnya, representasi permutasi dari itu) menjelaskan keberadaan rumus untuk menghitung akar <a href="/wiki/Persamaan_kuartik" title="Persamaan kuartik">persamaan kuartik</a> s dalam hal <a href="/w/index.php?title=Radikal_dari_suatu_grup_aljabar&action=edit&redlink=1" class="new" title="Radikal dari suatu grup aljabar (halaman belum tersedia)"> radikal</a>, sebagaimana ditetapkan oleh <a href="/w/index.php?title=Lodovico_Ferrari&action=edit&redlink=1" class="new" title="Lodovico Ferrari (halaman belum tersedia)">Lodovico Ferrari</a>: peta <span class="nowrap">S<sub>4</sub> → S<sub>3</sub></span> sesuai dengan kubik pemecah, dalam hal <a href="/wiki/Resolusi_Lagrange" class="mw-redirect" title="Resolusi Lagrange">resolusi Lagrange</a>. </p><p>Dalam konstruksi <a href="/w/index.php?title=Gelanggang_berhingga&action=edit&redlink=1" class="new" title="Gelanggang berhingga (halaman belum tersedia)">gelanggang berhingga</a>, delapan dari sebelas cincin dengan empat elemen memiliki empat kelompok Klein sebagai substruktur aditifnya. </p><p>Jika <b>R</b><sup>×</sup> menunjukkan kelompok perkalian dari non-nol real dan <b>R</b><sup>+</sup> kelompok perkalian <a href="/w/index.php?title=Riil_positif&action=edit&redlink=1" class="new" title="Riil positif (halaman belum tersedia)">riil positif</a>, <b>R</b><sup>×</sup> × <b>R</b><sup>×</sup> adalah <a href="/w/index.php?title=Grup_unit&action=edit&redlink=1" class="new" title="Grup unit (halaman belum tersedia)">grup unit</a> dari gelanggang <span class="nowrap"><b>R</b> × <b>R</b></span>, dan <span class="nowrap"><b>R</b><sup>+</sup> × <b>R</b><sup>+</sup></span> adalah subgrup dari <span class="nowrap"><b>R</b><sup>×</sup> × <b>R</b><sup>×</sup></span> (sebenarnya ini adalah <a href="/w/index.php?title=Komponen_identitas&action=edit&redlink=1" class="new" title="Komponen identitas (halaman belum tersedia)">komponen identitas</a> dari <span class="nowrap"><b>R</b><sup>×</sup> × <b>R</b><sup>×</sup></span>). <a href="/wiki/Grup_hasil_bagi" title="Grup hasil bagi">Grup hasil bagi</a> <span class="nowrap">(<b>R</b><sup>×</sup> × <b>R</b><sup>×</sup>) / (<b>R</b><sup>+</sup> × <b>R</b><sup>+</sup>)</span> isomorfik ke empat kelompok Klein. Dengan cara yang sama, kelompok unit <a href="/w/index.php?title=Bilangan_kompleks-pisah&action=edit&redlink=1" class="new" title="Bilangan kompleks-pisah (halaman belum tersedia)"> gelanggang bilangan kompleks pisah</a>, jika dibagi dengan komponen identitasnya, juga menghasilkan grup empat Klein. </p> <div class="mw-heading mw-heading2"><h2 id="Teori_grafik">Teori grafik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=5" title="Sunting bagian: Teori grafik" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=5" title="Sunting kode sumber bagian: Teori grafik"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Grafik_sederhana&action=edit&redlink=1" class="new" title="Grafik sederhana (halaman belum tersedia)"> sederhana</a> <a href="/w/index.php?title=Grafik_terhubung&action=edit&redlink=1" class="new" title="Grafik terhubung (halaman belum tersedia)">grafik terhubung</a> yang paling sederhana yang mengakui empat grup Klein sebagai <a href="/w/index.php?title=Grafik_automorfisme&action=edit&redlink=1" class="new" title="Grafik automorfisme (halaman belum tersedia)"> grup automorfisme</a> adalah <a href="/w/index.php?title=Grafik_berlian&action=edit&redlink=1" class="new" title="Grafik berlian (halaman belum tersedia)">grafik berlian</a> yang ditunjukkan di bawah ini. Ini juga merupakan grup automorfisme dari beberapa grafik lain yang lebih sederhana dalam arti memiliki lebih sedikit entitas. Ini termasuk grafik dengan empat simpul dan satu sisi, yang tetap sederhana tetapi kehilangan konektivitas, dan grafik dengan dua simpul yang dihubungkan satu sama lain oleh dua sisi, yang tetap terhubung tetapi kehilangan kesederhanaan. </p> <style data-mw-deduplicate="TemplateStyles:r15025838/mw-parser-output/.tmulti">.mw-parser-output .tmulti .thumbinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{text-align:left;background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .thumbcaption{text-align:center}}</style><div class="thumb tmulti tnone center"><div class="thumbinner" style="width:650px;max-width:650px"><div class="trow"><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Berkas:Diamond_graph.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Diamond_graph.svg/150px-Diamond_graph.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Diamond_graph.svg/225px-Diamond_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Diamond_graph.svg/300px-Diamond_graph.svg.png 2x" data-file-width="119" data-file-height="119" /></a></span></div></div><div class="tsingle" style="width:152px;max-width:152px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Berkas:Klein_4-Group_Graph.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Klein_4-Group_Graph.svg/150px-Klein_4-Group_Graph.svg.png" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Klein_4-Group_Graph.svg/225px-Klein_4-Group_Graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8f/Klein_4-Group_Graph.svg/300px-Klein_4-Group_Graph.svg.png 2x" data-file-width="256" data-file-height="256" /></a></span></div></div><div class="tsingle" style="width:340px;max-width:340px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/Berkas:Digon_graph.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Digon_graph.svg/338px-Digon_graph.svg.png" decoding="async" width="338" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Digon_graph.svg/507px-Digon_graph.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Digon_graph.svg/676px-Digon_graph.svg.png 2x" data-file-width="379" data-file-height="168" /></a></span></div></div></div></div></div> <div class="mw-heading mw-heading2"><h2 id="Musik">Musik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=6" title="Sunting bagian: Musik" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=6" title="Sunting kode sumber bagian: Musik"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dalam <a href="/wiki/Komposisi_musik" title="Komposisi musik">komposisi musik</a>, empat kelompok adalah kelompok dasar permutasi dalam <a href="/w/index.php?title=Teknik_dua_belas_nada&action=edit&redlink=1" class="new" title="Teknik dua belas nada (halaman belum tersedia)">teknik dua belas nada</a>. Dalam contoh itu tabel Cayley ditulis;<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <tbody><tr> <td>S</td> <td>I:</td> <td>R:</td> <td>RI: </td></tr> <tr> <td>I:</td> <td>S</td> <td>RI</td> <td>R </td></tr> <tr> <td>R:</td> <td>RI</td> <td>S</td> <td>I </td></tr> <tr> <td>RI:</td> <td>R</td> <td>I</td> <td>S </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Lihat_pula">Lihat pula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=7" title="Sunting bagian: Lihat pula" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=7" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Grup_angka_empat" class="mw-redirect" title="Grup angka empat">Grup angka empat</a></li> <li><a href="/wiki/Daftar_grup_kecil" title="Daftar grup kecil">Daftar grup kecil</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referensi">Referensi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=8" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=8" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r18833634">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><i>Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade</i> (Kuliah tentang ikosahedron dan solusi persamaan derajat kelima)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="/wiki/Milton_Babbitt" title="Milton Babbitt">Babbitt, Milton</a>. (1960) "Twelve-Tone Invariants as Compositional Determinants", <i>Musical Quarterly</i> 46(2):253 Edisi Khusus: Masalah Musik Modern: Seminar Princeton dalam Studi Musik Tingkat Lanjut (April): 246–59, <a href="/wiki/Oxford_University_Press" title="Oxford University Press">Oxford University Press</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Bacaan_lebih_lanjut">Bacaan lebih lanjut</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=9" title="Sunting bagian: Bacaan lebih lanjut" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=9" title="Sunting kode sumber bagian: Bacaan lebih lanjut"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>M. A. Armstrong (1988) <i>Groups and Symmetry</i>, <a href="/w/index.php?title=Springer_Verlag&action=edit&redlink=1" class="new" title="Springer Verlag (halaman belum tersedia)">Springer Verlag</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lDfcBPl9Cu4C&pg=PA53&dq=%22Klein%27s+group%22">page 53</a>.</li> <li>W. E. Barnes (1963) <i>Introduction to Abstract Algebra</i>, D.C. Heath & Co., page 20.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Pranala_luar">Pranala luar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Klein_empat_grup&veaction=edit&section=10" title="Sunting bagian: Pranala luar" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Klein_empat_grup&action=edit&section=10" title="Sunting kode sumber bagian: Pranala luar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Vierergruppe"><cite class="citation web"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Vierergruppe.html">"Vierergruppe"</a>. <i><a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Vierergruppe&rft.au=Weisstein%2C+Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FVierergruppe.html&rfr_id=info%3Asid%2Fid.wikipedia.org%3AKlein+empat+grup" class="Z3988"><span style="display:none;"> </span></span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Diperoleh dari "<a dir="ltr" href="https://id.wikipedia.org/w/index.php?title=Klein_empat_grup&oldid=24009583">https://id.wikipedia.org/w/index.php?title=Klein_empat_grup&oldid=24009583</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Istimewa:Daftar_kategori" title="Istimewa:Daftar kategori">Kategori</a>: <ul><li><a href="/wiki/Kategori:Grup_hingga" title="Kategori:Grup hingga">Grup hingga</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"> Halaman ini terakhir diubah pada 13 Agustus 2023, pukul 09.07.</li> <li id="footer-info-copyright">Teks tersedia di bawah <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Lisensi Atribusi-BerbagiSerupa Creative Commons</a>; ketentuan tambahan mungkin berlaku. 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