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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Masuk log</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Halaman penyunting yang telah keluar log <a href="/wiki/Bantuan:Pengantar" aria-label="Pelajari lebih lanjut tentang menyunting"><span>pelajari lebih lanjut</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Istimewa:Kontribusi_saya" title="Daftar suntingan yang dibuat dari alamat IP ini [y]" accesskey="y"><span>Kontribusi</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Istimewa:Pembicaraan_saya" title="Pembicaraan tentang suntingan dari alamat IP ini [n]" accesskey="n"><span>Pembicaraan</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Situs"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Daftar isi" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Daftar isi</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">pindah ke bilah sisi</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sembunyikan</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Awal</div> </a> </li> <li id="toc-Contoh" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Contoh"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Contoh</span> </div> </a> <button aria-controls="toc-Contoh-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Contoh</span> </button> <ul id="toc-Contoh-sublist" class="vector-toc-list"> <li id="toc-Grup_sederhana_hingga" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grup_sederhana_hingga"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Grup sederhana hingga</span> </div> </a> <ul id="toc-Grup_sederhana_hingga-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grup_sederhana_tak_terbatas" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grup_sederhana_tak_terbatas"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Grup sederhana tak terbatas</span> </div> </a> <ul id="toc-Grup_sederhana_tak_terbatas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Klasifikasi" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Klasifikasi"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Klasifikasi</span> </div> </a> <button aria-controls="toc-Klasifikasi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Klasifikasi</span> </button> <ul id="toc-Klasifikasi-sublist" class="vector-toc-list"> <li id="toc-Grup_sederhana_hingga_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Grup_sederhana_hingga_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Grup sederhana hingga</span> </div> </a> <ul id="toc-Grup_sederhana_hingga_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Struktur_grup_sederhana_berhingga" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Struktur_grup_sederhana_berhingga"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Struktur grup sederhana berhingga</span> </div> </a> <ul id="toc-Struktur_grup_sederhana_berhingga-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sejarah_untuk_kelompok_sederhana_hingga" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sejarah_untuk_kelompok_sederhana_hingga"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sejarah untuk kelompok sederhana hingga</span> </div> </a> <button aria-controls="toc-Sejarah_untuk_kelompok_sederhana_hingga-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Sejarah untuk kelompok sederhana hingga</span> </button> <ul id="toc-Sejarah_untuk_kelompok_sederhana_hingga-sublist" class="vector-toc-list"> <li id="toc-Konstruksi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Konstruksi"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Konstruksi</span> </div> </a> <ul id="toc-Konstruksi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Klasifikasi_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Klasifikasi_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Klasifikasi</span> </div> </a> <ul id="toc-Klasifikasi_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Tes_untuk_kesederhanaan" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tes_untuk_kesederhanaan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Tes untuk kesederhanaan</span> </div> </a> <ul id="toc-Tes_untuk_kesederhanaan-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">6</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">7</span> <span>Referensi</span> </div> </a> <button aria-controls="toc-Referensi-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Gulingkan subbagian Referensi</span> </button> <ul id="toc-Referensi-sublist" class="vector-toc-list"> <li id="toc-Catatan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Catatan"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Catatan</span> </div> </a> <ul id="toc-Catatan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Buku_teks" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Buku_teks"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.2</span> <span>Buku teks</span> </div> </a> <ul id="toc-Buku_teks-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dokumen" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dokumen"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.3</span> <span>Dokumen</span> </div> </a> <ul id="toc-Dokumen-sublist" class="vector-toc-list"> </ul> </li> </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">8</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 class="mw-body-header vector-page-titlebar"> <nav aria-label="Daftar isi" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Gulingkan daftar isi" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Gulingkan daftar isi</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Grup sederhana</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 24 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-24" 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">24 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_%D8%A8%D8%B3%D9%8A%D8%B7%D8%A9" title="زمرة بسيطة – Arab" lang="ar" hreflang="ar" data-title="زمرة بسيطة" 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_simple" title="Grup simple – Katalan" lang="ca" hreflang="ca" data-title="Grup simple" 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/Jednoduch%C3%A1_grupa" title="Jednoduchá grupa – Cheska" lang="cs" hreflang="cs" data-title="Jednoduchá grupa" data-language-autonym="Čeština" data-language-local-name="Cheska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Einfache_Gruppe_(Mathematik)" title="Einfache Gruppe (Mathematik) – Jerman" lang="de" hreflang="de" data-title="Einfache Gruppe (Mathematik)" 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/Simple_group" title="Simple group – Inggris" lang="en" hreflang="en" data-title="Simple 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_simple" title="Grupo simple – Spanyol" lang="es" hreflang="es" data-title="Grupo simple" 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_%D8%B3%D8%A7%D8%AF%D9%87" title="گروه ساده – Persia" lang="fa" hreflang="fa" data-title="گروه ساده" 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/Yksinkertainen_ryhm%C3%A4" title="Yksinkertainen ryhmä – Suomi" lang="fi" hreflang="fi" data-title="Yksinkertainen ryhmä" 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_simple" title="Groupe simple – Prancis" lang="fr" hreflang="fr" data-title="Groupe simple" 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%94_%D7%A4%D7%A9%D7%95%D7%98%D7%94" title="חבורה פשוטה – Ibrani" lang="he" hreflang="he" data-title="חבורה פשוטה" 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/Egyszer%C5%B1_csoport" title="Egyszerű csoport – Hungaria" lang="hu" hreflang="hu" data-title="Egyszerű csoport" data-language-autonym="Magyar" data-language-local-name="Hungaria" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_semplice" title="Gruppo semplice – Italia" lang="it" hreflang="it" data-title="Gruppo semplice" 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/%E5%8D%98%E7%B4%94%E7%BE%A4" title="単純群 – Jepang" lang="ja" hreflang="ja" data-title="単純群" 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/%EB%8B%A8%EC%88%9C%EA%B5%B0" title="단순군 – Korea" lang="ko" hreflang="ko" data-title="단순군" 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%B2%E0%B4%B3%E0%B4%BF%E0%B4%A4%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="ലളിതഗ്രൂപ്പ് – Malayalam" lang="ml" hreflang="ml" data-title="ലളിതഗ്രൂപ്പ്" 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/Enkelvoudige_groep" title="Enkelvoudige groep – Belanda" lang="nl" hreflang="nl" data-title="Enkelvoudige groep" 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_prosta" title="Grupa prosta – Polski" lang="pl" hreflang="pl" data-title="Grupa prosta" data-language-autonym="Polski" data-language-local-name="Polski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_simples" title="Grupo simples – Portugis" lang="pt" hreflang="pt" data-title="Grupo simples" 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/Grup_simplu" title="Grup simplu – Rumania" lang="ro" hreflang="ro" data-title="Grup simplu" 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%9F%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Простая группа – Rusia" lang="ru" hreflang="ru" data-title="Простая группа" 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/Enkel_grupp" title="Enkel grupp – Swedia" lang="sv" hreflang="sv" data-title="Enkel grupp" data-language-autonym="Svenska" data-language-local-name="Swedia" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D0%BE%D1%81%D1%82%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Проста група – Ukraina" lang="uk" hreflang="uk" data-title="Проста група" 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_%C4%91%C6%A1n" title="Nhóm đơn – Vietnam" lang="vi" hreflang="vi" data-title="Nhóm đơn" 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%8D%95%E7%BE%A4" title="单群 – Tionghoa" lang="zh" hreflang="zh" data-title="单群" 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/Q571124#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/Grup_sederhana" 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:Grup_sederhana" 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" 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href="/wiki/Grup_sederhana"><span>Baca</span></a></li><li id="ca-ve-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Grup_sederhana&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=Grup_sederhana&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=Grup_sederhana&action=history" title="Revisi sebelumnya dari halaman ini. 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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: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 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.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 class="mw-selflink selflink">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>grup sederhana</b> adalah sebuah nontrivial <a href="/wiki/Grup_(matematika)" title="Grup (matematika)">grup</a> yang hanya <a href="/wiki/Subgrup_normal" title="Subgrup normal">subgrup normal</a> adalah <a href="/w/index.php?title=Grup_trivial&action=edit&redlink=1" class="new" title="Grup trivial (halaman belum tersedia)">grup trivial</a> dan grup itu sendiri. Suatu grup yang tidak sederhana dapat dipecah menjadi dua grup yang lebih kecil, yaitu subgrup normal nontrivial dan <a href="/wiki/Grup_hasil_bagi" title="Grup hasil bagi">grup hasil bagi</a> yang sesuai. Proses ini dapat diulangi, dan untuk <a href="/wiki/Grup_terbatas" class="mw-redirect" title="Grup terbatas">grup terbatas</a> seseorang akhirnya sampai pada grup sederhana yang ditentukan secara unik, dengan <a href="/w/index.php?title=Teorema_Jordan%E2%80%93H%C3%B6lder&action=edit&redlink=1" class="new" title="Teorema Jordan–Hölder (halaman belum tersedia)">teorema Jordan–Hölder</a>. </p><p><a href="/wiki/Klasifikasi_grup_sederhana_hingga" title="Klasifikasi grup sederhana hingga">Klasifikasi grup sederhana hingga</a> yang lengkap, diselesaikan pada tahun 2004, merupakan tonggak penting dalam sejarah matematika. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Contoh">Contoh</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=1" title="Sunting bagian: Contoh" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=1" title="Sunting kode sumber bagian: Contoh"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Grup_sederhana_hingga">Grup sederhana hingga</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=2" title="Sunting bagian: Grup sederhana hingga" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=2" title="Sunting kode sumber bagian: Grup sederhana hingga"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Grup_siklik" title="Grup siklik">Grup siklik</a> <i>G</i> = <b>Z</b>/3<b>Z</b> dari <a href="/w/index.php?title=Kongruensi_kelas&action=edit&redlink=1" class="new" title="Kongruensi kelas (halaman belum tersedia)">kongruensi kelas</a> es <a href="/w/index.php?title=Operasi_modulo&action=edit&redlink=1" class="new" title="Operasi modulo (halaman belum tersedia)">modulo</a> 3 (lihat <a href="/wiki/Aritmatika_modular" class="mw-redirect" title="Aritmatika modular">aritmatika modular</a>) sederhana. Jika <i> H </i> adalah subgrup dari grup ini, nya <a href="/w/index.php?title=Urutan_(teori_grup)&action=edit&redlink=1" class="new" title="Urutan (teori grup) (halaman belum tersedia)">urutan</a> (jumlah elemen) harus menjadi <a href="/wiki/Pembagi" title="Pembagi">pembagi</a> dari urutan <i> G </i> yaitu 3. Karena 3 adalah bilangan prima, satu-satunya pembagi adalah 1 dan 3, jadi baik <i> H </i> adalah <i> G </i>, atau <i> H </i> adalah grup trivial. Di sisi lain, grup <i>G</i> = <b>Z</b>/12<b>Z</b> tidak sederhana. Himpunan <i> H </i> dari kelas-kelas kesesuaian dari 0, 4, dan 8 modulo 12 adalah subgrup berorde 3, dan ini adalah subkelompok normal karena setiap subkelompok dari <a href="/wiki/Grup_abelian" class="mw-redirect" title="Grup abelian">grup abelian</a> adalah normal. Demikian pula, grup aditif <b>Z</b> dari <a href="/wiki/Integer" class="mw-redirect" title="Integer">integer</a> s tidak sederhana; himpunan bilangan bulat genap adalah subgrup normal non-trivial yang tepat.<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> </p><p>Seseorang dapat menggunakan jenis penalaran yang sama untuk setiap grup abelian, untuk menyimpulkan bahwa satu-satunya grup abelian sederhana adalah grup siklik dari urutan <a href="/wiki/Bilangan_prima" title="Bilangan prima">prima</a>. Klasifikasi kelompok sederhana nonabelian jauh lebih sepele. Kelompok sederhana non abelian terkecil adalah <a href="/w/index.php?title=Grup_bergantian&action=edit&redlink=1" class="new" title="Grup bergantian (halaman belum tersedia)">grup bergantian</a> <i>A</i><sub>5</sub> dari orde 60, dan setiap grup orde 60 sederhana adalah <a href="/w/index.php?title=Isomorfisme_grup&action=edit&redlink=1" class="new" title="Isomorfisme grup (halaman belum tersedia)">isomorfis</a> pafa <i>A</i><sub>5</sub>.<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> Kelompok sederhana nonabelian terkecil kedua adalah kelompok linier khusus proyektif <a href="/w/index.php?title=PSL(2,7)&action=edit&redlink=1" class="new" title="PSL(2,7) (halaman belum tersedia)">PSL(2,7)</a> dengan orde 168, dan adalah mungkin untuk membuktikan bahwa setiap kelompok orde 168 sederhana isomorfik ke <a href="/w/index.php?title=PSL(2,7)&action=edit&redlink=1" class="new" title="PSL(2,7) (halaman belum tersedia)">PSL(2,7)</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Grup_sederhana_tak_terbatas">Grup sederhana tak terbatas</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=3" title="Sunting bagian: Grup sederhana tak terbatas" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=3" title="Sunting kode sumber bagian: Grup sederhana tak terbatas"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grup bergantian tak terbatas, yaitu grup permutasi yang didukung bahkan hingga bilangan bulat, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b6dfe5968776343496f22a0a90c8406065def1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="{\displaystyle A_{\infty }}"></span>. Grup ini dapat ditulis sebagai penyatuan yang meningkat dari grup sederhana hingga <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span> sehubungan dengan embedding standar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}\to A_{n+1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}\to A_{n+1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea334eb83fba418137ebce3f12921a3a97ebc986" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.285ex; height:2.509ex;" alt="{\displaystyle A_{n}\to A_{n+1}.}"></span> Keluarga contoh lain dari kelompok sederhana tak terbatas diberikan oleh <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} _{n}(F),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">P</mi> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {PSL} _{n}(F),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15d3ad74304a1c3af4ee3829ade78dbed7c8a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.744ex; height:2.843ex;" alt="{\displaystyle \mathrm {PSL} _{n}(F),}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> adalah bidang tak terbatas dan <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 2.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12579de3af09ac1e4dd0c0724536b2361760f498" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.302ex; height:2.343ex;" alt="{\displaystyle n\geq 2.}"></span> </p><p>Jauh lebih sulit untuk membangun grup sederhana tanpa batas yang <i> dihasilkan secara terbatas </i>. Hasil keberadaan pertama tidak eksplisit; hal ini disebabkan oleh <a href="/wiki/Graham_Higman" title="Graham Higman">Graham Higman</a> dan terdiri dari quotients sederhana dari <a href="/w/index.php?title=Grup_Higman&action=edit&redlink=1" class="new" title="Grup Higman (halaman belum tersedia)">grup Higman</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Contoh eksplisit, yang ternyata disajikan secara halus, termasuk <a href="/w/index.php?title=Gruo_Thompson&action=edit&redlink=1" class="new" title="Gruo Thompson (halaman belum tersedia)">gruo Thompson</a> <i> T </i> dan <i> V </i> yang tidak terbatas. Grup sederhana tak terbatas yang disajikan dengan sempurna <a href="/w/index.php?title=Torsi_(aljabar)&action=edit&redlink=1" class="new" title="Torsi (aljabar) (halaman belum tersedia)">bebas torsi</a> dibuat oleh Burger-Moze.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Klasifikasi">Klasifikasi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=4" title="Sunting bagian: Klasifikasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=4" title="Sunting kode sumber bagian: Klasifikasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Belum ada klasifikasi yang diketahui untuk kelompok sederhana umum (tak terbatas), dan klasifikasi semacam itu diharapkan tidak ada. </p> <div class="mw-heading mw-heading3"><h3 id="Grup_sederhana_hingga_2">Grup sederhana hingga</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=5" title="Sunting bagian: Grup sederhana hingga" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=5" title="Sunting kode sumber bagian: Grup sederhana hingga"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <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">Artikel utama: <a href="/w/index.php?title=Daftar_grup_sederhana_hingga&action=edit&redlink=1" class="new" title="Daftar grup sederhana hingga (halaman belum tersedia)">daftar grup sederhana hingga</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r18844875"><div role="note" class="hatnote navigation-not-searchable">Informasi lebih lanjut: <a href="/wiki/Klasifikasi_grup_sederhana_hingga" title="Klasifikasi grup sederhana hingga">Klasifikasi grup sederhana hingga</a></div> <p><a href="/w/index.php?title=Daftar_grup_sederhana_hingga&action=edit&redlink=1" class="new" title="Daftar grup sederhana hingga (halaman belum tersedia)">grup sederhana hingga</a> penting karena dalam arti tertentu mereka adalah "blok bangunan dasar" dari semua grup hingga, agak mirip dengan cara <a href="/wiki/Bilangan_prima" title="Bilangan prima">bilangan prima</a> adalah blok bangunan dasar dari <a href="/wiki/Bilangan_bulat" title="Bilangan bulat">bilangan bulat</a>. Hal ini diungkapkan oleh <a href="/w/index.php?title=Teorema_Jordan%E2%80%93H%C3%B6lder&action=edit&redlink=1" class="new" title="Teorema Jordan–Hölder (halaman belum tersedia)">Teorema Jordan–Hölder</a> yang menyatakan bahwa dua <a href="/w/index.php?title=Rangkaian_komposisi&action=edit&redlink=1" class="new" title="Rangkaian komposisi (halaman belum tersedia)">rangkaian komposisi</a> dari grup tertentu memiliki panjang yang sama dan faktor yang sama, <a href="/w/index.php?title=Hingga&action=edit&redlink=1" class="new" title="Hingga (halaman belum tersedia)">hingga</a> <a href="/wiki/Permutasi" title="Permutasi">permutasi</a> dan <a href="/wiki/Isomorfisme" title="Isomorfisme">isomorfisme</a>. Dalam upaya kolaboratif yang besar, <a href="/w/index.php?title=Klasifikasi_kelompok_sederhana_hingga&action=edit&redlink=1" class="new" title="Klasifikasi kelompok sederhana hingga (halaman belum tersedia)">klasifikasi kelompok sederhana hingga</a> dinyatakan diselesaikan pada tahun 1983 oleh <a href="/w/index.php?title=Daniel_Gorenstein&action=edit&redlink=1" class="new" title="Daniel Gorenstein (halaman belum tersedia)">Daniel Gorenstein</a>, meskipun beberapa masalah muncul (khususnya dalam klasifikasi <a href="/w/index.php?title=Grup_kuasithin&action=edit&redlink=1" class="new" title="Grup kuasithin (halaman belum tersedia)">grup kuasithin</a>, yang dipasang pada tahun 2004). </p><p>Secara singkat, kelompok sederhana hingga diklasifikasikan sebagai tergeletak dalam salah satu dari 18 keluarga, atau menjadi salah satu dari 26 pengecualian: </p> <ul><li>Z<sub><i>p</i></sub> – <a href="/wiki/Grup_siklik" title="Grup siklik">grup siklik</a> dari urutan utama</li> <li><i>A</i><sub><i>n</i></sub> - <a href="/w/index.php?title=Grup_bergantian&action=edit&redlink=1" class="new" title="Grup bergantian (halaman belum tersedia)">grup bergantian</a> untuk <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a752e15bfe1dac8d617d014a77c275bfd4af0d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 5}"></span> <dl><dd>Grup alternatif dapat dianggap sebagai grup jenis Lie di atas <a href="/w/index.php?title=Bidang_dengan_satu_elemen&action=edit&redlink=1" class="new" title="Bidang dengan satu elemen (halaman belum tersedia)">bidang dengan satu elemen</a>, yang menyatukan keluarga ini dengan yang berikutnya, dan dengan demikian semua keluarga dari kelompok terbatas sederhana non-abelian dapat dianggap sebagai tipe Lie.</dd></dl></li> <li>Satu dari 16 keluarga <a href="/w/index.php?title=Grup_jenis_Lie&action=edit&redlink=1" class="new" title="Grup jenis Lie (halaman belum tersedia)">grup jenis Lie</a> <dl><dd><a href="/w/index.php?title=Grup_Tits&action=edit&redlink=1" class="new" title="Grup Tits (halaman belum tersedia)">Grup Tits</a> secara umum dianggap dari bentuk ini, meskipun secara tegas itu bukan dari tipe Lie, melainkan indeks 2 dalam grup tipe Lie.</dd></dl></li> <li>Salah satu dari 26 pengecualian, <a href="/w/index.php?title=Grup_sporadis&action=edit&redlink=1" class="new" title="Grup sporadis (halaman belum tersedia)">grup sporadis</a>, 20 di antaranya adalah subkelompok atau <a href="/w/index.php?title=Sub-hasil_bagi&action=edit&redlink=1" class="new" title="Sub-hasil bagi (halaman belum tersedia)">sub-hasil bagi</a> dari <a href="/w/index.php?title=Grup_monster&action=edit&redlink=1" class="new" title="Grup monster (halaman belum tersedia)">grup monster</a> dan disebut sebagai "Keluarga Bahagia", sedangkan 6 sisanya disebut sebagai <a href="/w/index.php?title=Grup_paria&action=edit&redlink=1" class="new" title="Grup paria (halaman belum tersedia)">paria</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Struktur_grup_sederhana_berhingga">Struktur grup sederhana berhingga</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=6" title="Sunting bagian: Struktur grup sederhana berhingga" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=6" title="Sunting kode sumber bagian: Struktur grup sederhana berhingga"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/w/index.php?title=Teorema_Feit_%E2%80%93_Thompson&action=edit&redlink=1" class="new" title="Teorema Feit – Thompson (halaman belum tersedia)">teorema</a> dari <a href="/w/index.php?title=Walter_Feit&action=edit&redlink=1" class="new" title="Walter Feit (halaman belum tersedia)">Feit</a> dan <a href="/wiki/John_G._Thompson" title="John G. Thompson">Thompson</a> menyatakan bahwa setiap kelompok berorde ganjil adalah <a href="/w/index.php?title=Grup_solvabel&action=edit&redlink=1" class="new" title="Grup solvabel (halaman belum tersedia)">dapat dipecahkan</a>. Oleh karena itu, setiap kelompok sederhana hingga memiliki urutan genap kecuali jika itu adalah siklus orde utama. </p><p><a href="/w/index.php?title=Konjektur_Schreier&action=edit&redlink=1" class="new" title="Konjektur Schreier (halaman belum tersedia)">Konjektur Schreier</a> menegaskan bahwa grup <a href="/w/index.php?title=Automorfisme_luar&action=edit&redlink=1" class="new" title="Automorfisme luar (halaman belum tersedia)">automorfisme luar</a> dari setiap grup sederhana hingga dapat dipecahkan. Ini dapat dibuktikan dengan menggunakan teorema klasifikasi. </p> <div class="mw-heading mw-heading2"><h2 id="Sejarah_untuk_kelompok_sederhana_hingga">Sejarah untuk kelompok sederhana hingga</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=7" title="Sunting bagian: Sejarah untuk kelompok sederhana hingga" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=7" title="Sunting kode sumber bagian: Sejarah untuk kelompok sederhana hingga"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ada dua alur dalam sejarah kelompok sederhana hingga, penemuan dan konstruksi kelompok dan keluarga sederhana tertentu, yang berlangsung dari karya Galois pada tahun 1820-an hingga pembangunan Monster pada tahun 1981; dan bukti bahwa daftar ini lengkap, yang dimulai pada abad ke-19, paling signifikan terjadi pada 1955 hingga 1983 (ketika kemenangan pada awalnya diumumkan), tetapi secara umum hanya disetujui untuk menjadi final. Hingga 2010<sup class="plainlinks noexcerpt noprint asof-tag update" style="display:none;"><a class="external text" href="https://id.wikipedia.org/w/index.php?title=Grup_sederhana&action=edit">[update]</a></sup>, bekerja untuk meningkatkan bukti dan pemahaman terus berlanjut; Lihat (<a href="#CITEREFSilvestri1979">Silvestri 1979</a>) untuk sejarah abad ke-19 tentang kelompok sederhana. </p> <div class="mw-heading mw-heading3"><h3 id="Konstruksi">Konstruksi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=8" title="Sunting bagian: Konstruksi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=8" title="Sunting kode sumber bagian: Konstruksi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grup sederhana telah dipelajari setidaknya sejak awal <a href="/wiki/Teori_Galois" title="Teori Galois">teori Galois</a>, di mana <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> menyadari bahwa fakta bahwa <a href="/w/index.php?title=Kelompok_pengganti&action=edit&redlink=1" class="new" title="Kelompok pengganti (halaman belum tersedia)">kelompok pengganti</a> pada lima atau lebih adalah sederhana (dan karenanya tidak dapat dipecahkan), yang dibuktikannya pada tahun 1831. Galois juga membangun <a href="/w/index.php?title=Grup_linear_khusus_proyektif&action=edit&redlink=1" class="new" title="Grup linear khusus proyektif (halaman belum tersedia)">grup linear khusus proyektif</a> dari sebuah bidang di atas bidang berhingga prima, PSL(2,<i>p</i>), dan mengatakan bahwa mereka sederhana untuk <i> p </i> bukan 2 atau 3. Ini terkandung dalam surat terakhirnya kepada Chevalier,<sup id="cite_ref-chevalier-letter_7-0" class="reference"><a href="#cite_note-chevalier-letter-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> dan merupakan contoh berikutnya dari grup sederhana hingga.<sup id="cite_ref-raw_8-0" class="reference"><a href="#cite_note-raw-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>The next discoveries were by <a href="/wiki/Camille_Jordan" title="Camille Jordan">Camille Jordan</a> in 1870.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Jordan telah menemukan 4 famili dari grup matriks sederhana di atas <a href="/w/index.php?title=Bidang_hingga&action=edit&redlink=1" class="new" title="Bidang hingga (halaman belum tersedia)">bidang hingga</a> orde utama, yang sekarang dikenal sebagai <a href="/w/index.php?title=Grup_klasik&action=edit&redlink=1" class="new" title="Grup klasik (halaman belum tersedia)">grup klasik</a>. </p><p>Pada waktu yang hampir bersamaan, diperlihatkan bahwa sebuah keluarga terdiri dari lima kelompok, disebut <a href="/w/index.php?title=Grup_Mathieu&action=edit&redlink=1" class="new" title="Grup Mathieu (halaman belum tersedia)">grup Mathieu</a> dan pertama kali dijelaskan oleh <a href="/w/index.php?title=%C3%89mile_L%C3%A9onard_Mathieu&action=edit&redlink=1" class="new" title="Émile Léonard Mathieu (halaman belum tersedia)">Émile Léonard Mathieu</a> pada tahun 1861 dan 1873, juga sederhana. Karena kelima kelompok ini dibangun dengan metode yang tidak menghasilkan banyak kemungkinan yang tak terhingga, mereka disebut "<a href="/w/index.php?title=Grup_sporadis&action=edit&redlink=1" class="new" title="Grup sporadis (halaman belum tersedia)">sporadis</a>" oleh <a href="/wiki/William_Burnside" title="William Burnside">William Burnside</a> dalam buku teksnya tahun 1897. </p><p>Kemudian hasil Jordan pada kelompok klasik digeneralisasikan ke bidang terbatas sewenang-wenang oleh <a href="/w/index.php?title=Leonard_Dickson&action=edit&redlink=1" class="new" title="Leonard Dickson (halaman belum tersedia)">Leonard Dickson</a>, mengikuti klasifikasi <a href="/w/index.php?title=Aljabar_Lie_sederhana_kompleks&action=edit&redlink=1" class="new" title="Aljabar Lie sederhana kompleks (halaman belum tersedia)">aljabar Lie sederhana kompleks</a> berdasarkan <a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a>. Dickson juga membangun grup pengecualian tipe G<sub>2</sub> dan <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> juga, tapi bukan tipe F<sub>4</sub>, E<sub>7</sub>, atau E<sub>8</sub> (<a href="#CITEREFWilson2009">Wilson 2009</a>, hlm. 2). Pada 1950-an pekerjaan kelompok tipe Lie dilanjutkan, dengan <a href="/w/index.php?title=Claude_Chevalley&action=edit&redlink=1" class="new" title="Claude Chevalley (halaman belum tersedia)">Claude Chevalley</a> memberikan konstruksi seragam dari kelompok klasik dan kelompok jenis luar biasa dalam kertas 1955. Ini menghilangkan kelompok tertentu yang diketahui (kelompok kesatuan proyektif), yang diperoleh dengan "memutar" konstruksi Chevalley. Kelompok tipe Lie yang tersisa diproduksi oleh Steinberg, Tits, dan Herzig (yang memproduseri <sup>3</sup><i>D</i><sub>4</sub>(<i>q</i>) and <sup>2</sup><i>E</i><sub>6</sub>(<i>q</i>)) dan oleh Suzuki dan Ree (<a href="/w/index.php?title=Grup_Suzuki%E2%80%93Ree&action=edit&redlink=1" class="new" title="Grup Suzuki–Ree (halaman belum tersedia)">grup Suzuki–Ree</a>). </p><p>Grup ini (grup tipe Lie, bersama dengan kelompok siklik, kelompok bergantian, dan lima kelompok Mathieu yang luar biasa) diyakini sebagai daftar lengkap, tetapi setelah jeda hampir satu abad sejak karya Mathieu, pada tahun 1964 <a href="/w/index.php?title=Gruo_Janko&action=edit&redlink=1" class="new" title="Gruo Janko (halaman belum tersedia)">gruo Janko</a> pertama ditemukan, dan sisa 20 grup sporadis ditemukan atau diduga pada tahun 1965–1975, berpuncak pada tahun 1981, ketika <a href="/w/index.php?title=Robert_Griess&action=edit&redlink=1" class="new" title="Robert Griess (halaman belum tersedia)">Robert Griess</a> mengumumkan bahwa ia telah membangun "<a href="/w/index.php?title=Grup_Monster&action=edit&redlink=1" class="new" title="Grup Monster (halaman belum tersedia)">grup Monster</a>" milik <a href="/w/index.php?title=Bernd_Fischer_(matematikawan)&action=edit&redlink=1" class="new" title="Bernd Fischer (matematikawan) (halaman belum tersedia)">Bernd Fischer</a> ". Monster adalah grup sederhana sporadis terbesar yang memiliki urutan 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. Monster memiliki representasi 196.883 dimensi yang setia dalam <a href="/w/index.php?title=Aljabar_Griess&action=edit&redlink=1" class="new" title="Aljabar Griess (halaman belum tersedia)">Aljabar Griess</a> dimensi 196.884, yang berarti bahwa setiap elemen Monster dapat diekspresikan sebagai matriks 196.883 x 196.883. </p> <div class="mw-heading mw-heading3"><h3 id="Klasifikasi_2">Klasifikasi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=9" title="Sunting bagian: Klasifikasi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=9" title="Sunting kode sumber bagian: Klasifikasi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Klasifikasi lengkap secara umum diterima sebagai dimulai dengan <a href="/w/index.php?title=Teorema_Feit%E2%80%93Thompson&action=edit&redlink=1" class="new" title="Teorema Feit–Thompson (halaman belum tersedia)">Teorema Feit–Thompson</a> tahun 1962/63, sebagian besar berlangsung hingga tahun 1983, tetapi baru selesai pada tahun 2004. </p><p>Segera setelah pembangunan Monster pada tahun 1981, menjadi bukti, berjumlah lebih dari 10.000 halaman, asalkan ahli teori grup telah berhasil <a href="/w/index.php?title=Daftar_grup_sederhana_hingga&action=edit&redlink=1" class="new" title="Daftar grup sederhana hingga (halaman belum tersedia)">mendaftar semua grup sederhana hingga</a>, dengan kemenangan diumumkan pada tahun 1983 oleh Daniel Gorenstein. Ini terlalu dini, beberapa celah kemudian ditemukan, terutama dalam klasifikasi <a href="/w/index.php?title=Grup_kuasithin&action=edit&redlink=1" class="new" title="Grup kuasithin (halaman belum tersedia)">grup kuasithin</a>, yang akhirnya diganti pada tahun 2004 oleh klasifikasi grup quasithin 1.300 halaman, yang sekarang secara umum diterima sebagai lengkap. </p> <div class="mw-heading mw-heading2"><h2 id="Tes_untuk_kesederhanaan">Tes untuk kesederhanaan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=10" title="Sunting bagian: Tes untuk kesederhanaan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=10" title="Sunting kode sumber bagian: Tes untuk kesederhanaan"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><a href="/wiki/Teorema_Sylow#Contoh_aplikasi" title="Teorema Sylow">Pengujian Sylow</a></i>: Misalkan <i> n </i> adalah bilangan bulat positif yang bukan prima, dan misalkan <i> p </i> menjadi pembagi prima dari <i> n </i>. Jika 1 adalah satu-satunya pembagi dari <i> n </i> yang sama dengan 1 modulo p, maka tidak ada grup orde sederhana <i> n </i>. </p><p>Bukti: Jika <i> n </i> adalah kekuatan-prima, maka segrup urutan <i> n </i> memiliki nontrivial <a href="/w/index.php?title=Pusat_(teori_grup)&action=edit&redlink=1" class="new" title="Pusat (teori grup) (halaman belum tersedia)">pusat</a><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> dan, oleh karena itu, tidaklah sederhana. Jika <i> n </i> bukan pangkat utama, maka setiap subkelompok Sylow adalah tepat, dan, menurut <a href="/wiki/Teorema_Sylow" title="Teorema Sylow">Teorema Ketiga Sylow</a>, kita tahu bahwa jumlah subgrup p Sylow dari kelompok orde <i> n </i> sama dengan 1 modulo <i> p </i> dan membagi <i> n </i>. Karena 1 adalah satu-satunya bilangan tersebut, subgrup p Sylow unik, dan oleh karena itu normal. Karena ini adalah subkelompok non-identitas yang tepat, kelompok ini tidak sederhana. </p><p><i>Burnside</i>: Grup sederhana hingga non-Abelian memiliki urutan yang habis dibagi oleh setidaknya tiga bilangan prima yang berbeda. Ini mengikuti dari <a href="/w/index.php?title=Teorema_Burnside&action=edit&redlink=1" class="new" title="Teorema Burnside (halaman belum tersedia)">Teorema p-q Burnside</a>. </p> <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=Grup_sederhana&veaction=edit&section=11" 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=Grup_sederhana&action=edit&section=11" title="Sunting kode sumber bagian: Lihat pula"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/w/index.php?title=Grup_hampir_sederhana&action=edit&redlink=1" class="new" title="Grup hampir sederhana (halaman belum tersedia)">Grup hampir sederhana</a></li> <li><a href="/w/index.php?title=Grup_dengan_karakteristik_sederhana&action=edit&redlink=1" class="new" title="Grup dengan karakteristik sederhana (halaman belum tersedia)">Grup dengan karakteristik sederhana</a></li> <li>Grup sederhana</li> <li><a href="/w/index.php?title=Grup_semi-sederhana&action=edit&redlink=1" class="new" title="Grup semi-sederhana (halaman belum tersedia)">Grup semi-sederhana</a></li> <li><a href="/w/index.php?title=Daftar_grup_sederhana_hingga&action=edit&redlink=1" class="new" title="Daftar grup sederhana hingga (halaman belum tersedia)">Daftar grup sederhana hingga</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=Grup_sederhana&veaction=edit&section=12" title="Sunting bagian: Referensi" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=12" title="Sunting kode sumber bagian: Referensi"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Catatan">Catatan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=13" title="Sunting bagian: Catatan" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=13" title="Sunting kode sumber bagian: Catatan"><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">Knapp (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=KVeXG163BggC&pg=PA170&dq=%22Z+is+not+simple%2C+having+the+nontrivial+subgroup+2Z%22">p. 170</a></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">Rotman (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lYrsiaHSHKcC&pg=PA226&dq=%22simple+groups+of+order+60+are+isomorphic%22">p. 226</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Rotman (1995), p. 281</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Smith & Tabachnikova (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DD0TW28WjfQC&pg=PA144&dq=%22any+two+simple+groups+of+order+168+are+isomorphic%22">p. 144</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFHigman1951" class="citation"><a href="/wiki/Graham_Higman" title="Graham Higman">Higman, Graham</a> (1951), "A finitely generated infinite simple group", <i>Journal of the London Mathematical Society</i>, Second Series, <b>26</b> (1): 61–64, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fjlms%2Fs1-26.1.59">10.1112/jlms/s1-26.1.59</a>, <a href="/wiki/International_Standard_Serial_Number" class="mw-redirect" title="International Standard Serial Number">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0024-6107">0024-6107</a>, <a href="/wiki/Mathematical_Reviews" title="Mathematical Reviews">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0038348">0038348</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+London+Mathematical+Society&rft.atitle=A+finitely+generated+infinite+simple+group&rft.volume=26&rft.issue=1&rft.pages=61-64&rft.date=1951&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0038348&rft.issn=0024-6107&rft_id=info%3Adoi%2F10.1112%2Fjlms%2Fs1-26.1.59&rft.aulast=Higman&rft.aufirst=Graham&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite class="citation journal">Burger, M.; Mozes, S. (2000). "Lattices in product of trees". <i>Publ. Math. IHES</i>. <b>92</b>: 151–194. <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02698916">10.1007/bf02698916</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publ.+Math.+IHES&rft.atitle=Lattices+in+product+of+trees&rft.volume=92&rft.pages=151-194&rft.date=2000&rft_id=info%3Adoi%2F10.1007%2Fbf02698916&rft.aulast=Burger&rft.aufirst=M.&rft.au=Mozes%2C+S.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-chevalier-letter-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-chevalier-letter_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFGalois1846" class="citation">Galois, Évariste (1846), <a rel="nofollow" class="external text" href="http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-16390&I=416&M=tdm">"Lettre de Galois à M. Auguste Chevalier"</a>, <i><a href="/wiki/Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es" title="Journal de Mathématiques Pures et Appliquées">Journal de Mathématiques Pures et Appliquées</a></i>, <b>XI</b>: 408–415, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221126151915/http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-16390&I=416&M=tdm">diarsipkan</a> dari versi asli tanggal 2022-11-26<span class="reference-accessdate">, diakses tanggal <span class="nowrap">2009-02-04</span></span>, PSL(2,<i>p</i>) dan kesederhanaan dibahas pada hal. 411; tindakan luar biasa pada 5, 7, atau 11 poin yang dibahas pada hlm. 411–412; GL(<i>ν</i>,<i>p</i>) dibahas di hal. 410</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+de+Math%C3%A9matiques+Pures+et+Appliqu%C3%A9es&rft.atitle=Lettre+de+Galois+%C3%A0+M.+Auguste+Chevalier&rft.volume=XI&rft.pages=408-415&rft.date=1846&rft.aulast=Galois&rft.aufirst=%C3%89variste&rft_id=http%3A%2F%2Fvisualiseur.bnf.fr%2FCadresFenetre%3FO%3DNUMM-16390%26I%3D416%26M%3Dtdm&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-raw-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-raw_8-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWilson2006" class="citation"><a href="/w/index.php?title=Robert_Arnott_Wilson&action=edit&redlink=1" class="new" title="Robert Arnott Wilson (halaman belum tersedia)">Wilson, Robert</a> (October 31, 2006), <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs_files/intro.ps">"Chapter 1: Introduction"</a>, <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs.html"><i>The finite simple groups</i></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110522121819/http://www.maths.qmul.ac.uk/~raw/fsgs.html">diarsipkan</a> dari versi asli tanggal 2011-05-22<span class="reference-accessdate">, diakses tanggal <span class="nowrap">2020-12-12</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapter+1%3A+Introduction&rft.btitle=The+finite+simple+groups&rft.date=2006-10-31&rft.aulast=Wilson&rft.aufirst=Robert&rft_id=http%3A%2F%2Fwww.maths.qmul.ac.uk%2F~raw%2Ffsgs_files%2Fintro.ps&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFJordan1870" class="citation"><a href="/wiki/Camille_Jordan" title="Camille Jordan">Jordan, Camille</a> (1870), <i><a href="/w/index.php?title=List_of_important_publications_in_mathematics&action=edit&redlink=1" class="new" title="List of important publications in mathematics (halaman belum tersedia)">Traité des substitutions et des équations algébriques</a></i></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Trait%C3%A9+des+substitutions+et+des+%C3%A9quations+alg%C3%A9briques&rft.date=1870&rft.aulast=Jordan&rft.aufirst=Camille&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Lihat bukti di <a href="/wiki/Grup-p" title="Grup-p">grup-p</a>, misalnya.</span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Buku_teks">Buku teks</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=14" title="Sunting bagian: Buku teks" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=14" title="Sunting kode sumber bagian: Buku teks"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r21093234">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}</style><div class="refbegin" style=""> <ul><li><cite id="CITEREFWilson2009" class="citation"><a href="/w/index.php?title=Robert_Arnott_Wilson&action=edit&redlink=1" class="new" title="Robert Arnott Wilson (halaman belum tersedia)">Wilson, Robert A.</a> (2009), <i>The finite simple groups</i>, <a href="/w/index.php?title=Graduate_Texts_in_Mathematics&action=edit&redlink=1" class="new" title="Graduate Texts in Mathematics (halaman belum tersedia)">Graduate Texts in Mathematics</a> 251, <b>251</b>, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-84800-988-2">10.1007/978-1-84800-988-2</a>, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-1-84800-987-5" title="Istimewa:Sumber buku/978-1-84800-987-5">978-1-84800-987-5</a>, <a href="/w/index.php?title=Zentralblatt_MATH&action=edit&redlink=1" class="new" title="Zentralblatt MATH (halaman belum tersedia)">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:1203.20012">1203.20012</a>, <a rel="nofollow" class="external text" href="http://www.maths.qmul.ac.uk/~raw/fsgs.html">2007 preprint</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+finite+simple+groups&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics+251&rft.pub=Springer-Verlag&rft.date=2009&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1203.20012&rft_id=info%3Adoi%2F10.1007%2F978-1-84800-988-2&rft.isbn=978-1-84800-987-5&rft.aulast=Wilson&rft.aufirst=Robert+A.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFBurnside1897" class="citation"><a href="/wiki/William_Burnside" title="William Burnside">Burnside, William</a> (1897), <i>Theory of groups of finite order</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Theory+of+groups+of+finite+order&rft.pub=Cambridge+University+Press&rft.date=1897&rft.aulast=Burnside&rft.aufirst=William&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></li></ul> </div> <ul><li><cite id="CITEREFKnapp2006" class="citation">Knapp, Anthony W. (2006), <i>Basic algebra</i>, Springer, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-8176-3248-9" title="Istimewa:Sumber buku/978-0-8176-3248-9">978-0-8176-3248-9</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.pub=Springer&rft.date=2006&rft.isbn=978-0-8176-3248-9&rft.aulast=Knapp&rft.aufirst=Anthony+W.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFRotman1995" class="citation">Rotman, Joseph J. (1995), <i>An introduction to the theory of groups</i>, Graduate texts in mathematics, <b>148</b>, Springer, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-0-387-94285-8" title="Istimewa:Sumber buku/978-0-387-94285-8">978-0-387-94285-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+the+theory+of+groups&rft.series=Graduate+texts+in+mathematics&rft.pub=Springer&rft.date=1995&rft.isbn=978-0-387-94285-8&rft.aulast=Rotman&rft.aufirst=Joseph+J.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite id="CITEREFSmithTabachnikova2000" class="citation">Smith, Geoff; Tabachnikova, Olga (2000), <i>Topics in group theory</i>, Springer undergraduate mathematics series (edisi ke-2), Springer, <a href="/wiki/International_Standard_Book_Number" class="mw-redirect" title="International Standard Book Number">ISBN</a> <a href="/wiki/Istimewa:Sumber_buku/978-1-85233-235-8" title="Istimewa:Sumber buku/978-1-85233-235-8">978-1-85233-235-8</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topics+in+group+theory&rft.series=Springer+undergraduate+mathematics+series&rft.edition=2&rft.pub=Springer&rft.date=2000&rft.isbn=978-1-85233-235-8&rft.aulast=Smith&rft.aufirst=Geoff&rft.au=Tabachnikova%2C+Olga&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Dokumen">Dokumen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Grup_sederhana&veaction=edit&section=15" title="Sunting bagian: Dokumen" class="mw-editsection-visualeditor"><span>sunting</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Grup_sederhana&action=edit&section=15" title="Sunting kode sumber bagian: Dokumen"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r21093234"><div class="refbegin" style=""> <ul><li><cite id="CITEREFSilvestri1979" class="citation">Silvestri, R. (September 1979), "Simple groups of finite order in the nineteenth century", <i>Archive for History of Exact Sciences</i>, <b>20</b> (3–4): 313–356, <a href="/wiki/Digital_object_identifier" class="mw-redirect" title="Digital object identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00327738">10.1007/BF00327738</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Archive+for+History+of+Exact+Sciences&rft.atitle=Simple+groups+of+finite+order+in+the+nineteenth+century&rft.volume=20&rft.issue=3%E2%80%934&rft.pages=313-356&rft.date=1979-09&rft_id=info%3Adoi%2F10.1007%2FBF00327738&rft.aulast=Silvestri&rft.aufirst=R.&rfr_id=info%3Asid%2Fid.wikipedia.org%3AGrup+sederhana" class="Z3988"><span style="display:none;"> </span></span></li></ul> </div> <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=Grup_sederhana&veaction=edit&section=16" 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=Grup_sederhana&action=edit&section=16" title="Sunting kode sumber bagian: Pranala luar"><span>sunting sumber</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/?op=getobj&from=objects&id=3569">The alternating group A_n is simple</a>, <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath.org</a>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" 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=Grup_sederhana&oldid=23984154">https://id.wikipedia.org/w/index.php?title=Grup_sederhana&oldid=23984154</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="/w/index.php?title=Kategori:Articles_containing_potentially_dated_statements_from_2010&action=edit&redlink=1" class="new" title="Kategori:Articles containing potentially dated statements from 2010 (halaman belum tersedia)">Articles containing potentially dated statements from 2010</a></li><li><a href="/w/index.php?title=Kategori:Sifat_grup&action=edit&redlink=1" class="new" title="Kategori:Sifat grup (halaman belum tersedia)">Sifat grup</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Kategori tersembunyi: <ul><li><a href="/wiki/Kategori:Artikel_dengan_parameter_tanggal_yang_tidak_valid_pada_templat" title="Kategori:Artikel dengan parameter tanggal yang tidak valid pada templat">Artikel dengan parameter tanggal yang tidak valid pada templat</a></li><li><a href="/wiki/Kategori:All_articles_containing_potentially_dated_statements" title="Kategori:All articles containing potentially dated statements">All articles containing potentially dated statements</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 9 Agustus 2023, pukul 15.41.</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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