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Tilgængelig på 56 sprog" > <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-56" 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">56 sprog</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Abelse_groep" title="Abelse groep – afrikaans" lang="af" hreflang="af" data-title="Abelse groep" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_%D8%A2%D8%A8%D9%84%D9%8A%D8%A9" title="زمرة آبلية – arabisk" lang="ar" hreflang="ar" data-title="زمرة آبلية" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Grupu_abelianu" title="Grupu abelianu – asturisk" lang="ast" hreflang="ast" data-title="Grupu abelianu" data-language-autonym="Asturianu" data-language-local-name="asturisk" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Abel_ortalamalar%C4%B1" title="Abel ortalamaları – aserbajdsjansk" lang="az" hreflang="az" data-title="Abel ortalamaları" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D1%8C_%D1%82%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D3%A9" title="Абель төркөмө – bashkir" lang="ba" hreflang="ba" data-title="Абель төркөмө" data-language-autonym="Башҡортса" data-language-local-name="bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Абелева група – belarusisk" lang="be" hreflang="be" data-title="Абелева група" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D0%BE%D0%B2%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Абелова група – bulgarsk" lang="bg" hreflang="bg" data-title="Абелова група" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%AC%E0%A7%87%E0%A6%B2%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A7%81%E0%A6%AA" title="আবেলীয় গ্রুপ – bengali" lang="bn" hreflang="bn" data-title="আবেলীয় গ্রুপ" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_abeli%C3%A0" title="Grup abelià – catalansk" lang="ca" hreflang="ca" data-title="Grup abelià" data-language-autonym="Català" data-language-local-name="catalansk" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%88%D9%BE%DB%8C_%D8%A6%D8%A7%DA%B5%D9%88%DA%AF%DB%86%DA%95%DB%8C" title="گرووپی ئاڵوگۆڕی – sorani" lang="ckb" hreflang="ckb" data-title="گرووپی ئاڵوگۆڕی" data-language-autonym="کوردی" data-language-local-name="sorani" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Abelova_grupa" title="Abelova grupa – tjekkisk" lang="cs" hreflang="cs" data-title="Abelova grupa" data-language-autonym="Čeština" data-language-local-name="tjekkisk" 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/Abelsche_Gruppe" title="Abelsche Gruppe – tysk" lang="de" hreflang="de" data-title="Abelsche Gruppe" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%B2%CE%B5%CE%BB%CE%B9%CE%B1%CE%BD%CE%AE_%CE%BF%CE%BC%CE%AC%CE%B4%CE%B1" title="Αβελιανή ομάδα – græsk" lang="el" hreflang="el" data-title="Αβελιανή ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="græsk" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Abelian_group" title="Abelian group – engelsk" lang="en" hreflang="en" data-title="Abelian group" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Komuta_grupo" title="Komuta grupo – esperanto" lang="eo" hreflang="eo" data-title="Komuta grupo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_abeliano" title="Grupo abeliano – spansk" lang="es" hreflang="es" data-title="Grupo abeliano" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Abeli_r%C3%BChm" title="Abeli rühm – estisk" lang="et" hreflang="et" data-title="Abeli rühm" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde_abeldar" title="Talde abeldar – baskisk" lang="eu" hreflang="eu" data-title="Talde abeldar" data-language-autonym="Euskara" data-language-local-name="baskisk" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_%D8%A2%D8%A8%D9%84%DB%8C" title="گروه آبلی – persisk" lang="fa" hreflang="fa" data-title="گروه آبلی" data-language-autonym="فارسی" data-language-local-name="persisk" 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/Abelin_ryhm%C3%A4" title="Abelin ryhmä – finsk" lang="fi" hreflang="fi" data-title="Abelin ryhmä" data-language-autonym="Suomi" data-language-local-name="finsk" 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_ab%C3%A9lien" title="Groupe abélien – fransk" lang="fr" hreflang="fr" data-title="Groupe abélien" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Grupo_abeliano" title="Grupo abeliano – galicisk" lang="gl" hreflang="gl" data-title="Grupo abeliano" data-language-autonym="Galego" data-language-local-name="galicisk" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%90%D7%91%D7%9C%D7%99%D7%AA" title="חבורה אבלית – hebraisk" lang="he" hreflang="he" data-title="חבורה אבלית" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Abelova_grupa" title="Abelova grupa – kroatisk" lang="hr" hreflang="hr" data-title="Abelova grupa" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Abel-csoport" title="Abel-csoport – ungarsk" lang="hu" hreflang="hu" data-title="Abel-csoport" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B1%D5%A2%D5%A5%D5%AC%D5%B5%D5%A1%D5%B6_%D5%AD%D5%B8%D6%82%D5%B4%D5%A2" title="Աբելյան խումբ – armensk" lang="hy" hreflang="hy" data-title="Աբելյան խումբ" data-language-autonym="Հայերեն" data-language-local-name="armensk" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_abelian" title="Gruppo abelian – interlingua" lang="ia" hreflang="ia" data-title="Gruppo abelian" data-language-autonym="Interlingua" data-language-local-name="interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_Abelian" title="Grup Abelian – indonesisk" lang="id" hreflang="id" data-title="Grup Abelian" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_abeliano" title="Gruppo abeliano – italiensk" lang="it" hreflang="it" data-title="Gruppo abeliano" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A2%E3%83%BC%E3%83%99%E3%83%AB%E7%BE%A4" title="アーベル群 – japansk" lang="ja" hreflang="ja" data-title="アーベル群" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Abel_topar%C4%B1" title="Abel toparı – karakalpakisk" lang="kaa" hreflang="kaa" data-title="Abel toparı" data-language-autonym="Qaraqalpaqsha" data-language-local-name="karakalpakisk" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%EB%B2%A8_%EA%B5%B0" title="아벨 군 – koreansk" lang="ko" hreflang="ko" data-title="아벨 군" data-language-autonym="한국어" data-language-local-name="koreansk" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Caterva_Abeliana" title="Caterva Abeliana – latin" lang="la" hreflang="la" data-title="Caterva Abeliana" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8D%E0%B4%B0%E0%B4%AE%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="ക്രമഗ്രൂപ്പ് – malayalam" lang="ml" hreflang="ml" data-title="ക്രമഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kumpulan_Abel" title="Kumpulan Abel – malajisk" lang="ms" hreflang="ms" data-title="Kumpulan Abel" data-language-autonym="Bahasa Melayu" data-language-local-name="malajisk" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Abelse_groep" title="Abelse groep – nederlandsk" lang="nl" hreflang="nl" data-title="Abelse groep" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Abelsk_gruppe" title="Abelsk gruppe – nynorsk" lang="nn" hreflang="nn" data-title="Abelsk gruppe" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Abelsk_gruppe" title="Abelsk gruppe – bokmål" lang="nb" hreflang="nb" data-title="Abelsk gruppe" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Abelan_grupe" title="Abelan grupe – Novial" lang="nov" hreflang="nov" data-title="Abelan grupe" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_przemienna" title="Grupa przemienna – polsk" lang="pl" hreflang="pl" data-title="Grupa przemienna" data-language-autonym="Polski" data-language-local-name="polsk" 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_abeliano" title="Grupo abeliano – portugisisk" lang="pt" hreflang="pt" data-title="Grupo abeliano" data-language-autonym="Português" data-language-local-name="portugisisk" 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_abelian" title="Grup abelian – rumænsk" lang="ro" hreflang="ro" data-title="Grup abelian" data-language-autonym="Română" data-language-local-name="rumænsk" 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%90%D0%B1%D0%B5%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="Абелева группа – russisk" lang="ru" hreflang="ru" data-title="Абелева группа" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Abelova_grupa" title="Abelova grupa – serbokroatisk" lang="sh" hreflang="sh" data-title="Abelova grupa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Abelian_group" title="Abelian group – Simple English" lang="en-simple" hreflang="en-simple" data-title="Abelian group" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Abelovsk%C3%A1_grupa" title="Abelovská grupa – slovakisk" lang="sk" hreflang="sk" data-title="Abelovská grupa" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Abelova_grupa" title="Abelova grupa – slovensk" lang="sl" hreflang="sl" data-title="Abelova grupa" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D0%BE%D0%B2%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Абелова група – serbisk" lang="sr" hreflang="sr" data-title="Абелова група" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Abelsk_grupp" title="Abelsk grupp – svensk" lang="sv" hreflang="sv" data-title="Abelsk grupp" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B0%E0%AE%BF%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="பரிமாற்றுக் குலம் – tamil" lang="ta" hreflang="ta" data-title="பரிமாற்றுக் குலம்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Abel_grubu" title="Abel grubu – tyrkisk" lang="tr" hreflang="tr" data-title="Abel grubu" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%B1%D0%B5%D0%BB%D0%B5%D0%B2%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Абелева група – ukrainsk" lang="uk" hreflang="uk" data-title="Абелева група" data-language-autonym="Українська" data-language-local-name="ukrainsk" 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_giao_ho%C3%A1n" title="Nhóm giao hoán – vietnamesisk" lang="vi" hreflang="vi" data-title="Nhóm giao hoán" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" 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/%E9%98%BF%E8%B4%9D%E5%B0%94%E7%BE%A4" title="阿贝尔群 – kinesisk" lang="zh" hreflang="zh" data-title="阿贝尔群" data-language-autonym="中文" data-language-local-name="kinesisk" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BA%A4%E6%8F%9B%E7%BE%A4" title="交換群 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="交換群" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%98%BF%E8%B2%9D%E7%88%BE%E7%BE%A3" title="阿貝爾羣 – kantonesisk" lang="yue" hreflang="yue" data-title="阿貝爾羣" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</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/Q181296#sitelinks-wikipedia" title="Redigér sproglinks" class="wbc-editpage">Redigér links</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="Navnerum"> <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/Abelsk_gruppe" title="Se indholdssiden [c]" accesskey="c"><span>Artikel</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Diskussion:Abelsk_gruppe&action=edit&redlink=1" rel="discussion" class="new" title="Diskussion om indholdet på siden (ikke skrevet endnu) [t]" accesskey="t"><span>Diskussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Ændr sprogvariant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" 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accesskey="v"><span>Redigér</span></a></li><li id="ca-edit" class="collapsible vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&action=edit" title="Rediger kildekoden for denne side [e]" accesskey="e"><span>Rediger kildekode</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&action=history" title="Tidligere versioner af denne side [h]" accesskey="h"><span>Se historik</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Sideværktøjer"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Værktøjer" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Værktøjer</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Værktøjer</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">flyt til sidebjælken</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">skjul</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Flere muligheder" > <div class="vector-menu-heading"> Handlinger </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Abelsk_gruppe"><span>Læs</span></a></li><li id="ca-more-ve-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&veaction=edit" title="Redigér denne side [v]" accesskey="v"><span>Redigér</span></a></li><li id="ca-more-edit" class="collapsible vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&action=edit" title="Rediger kildekoden for denne side [e]" accesskey="e"><span>Rediger kildekode</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&action=history"><span>Se historik</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Generelt </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speciel:Hvad_linker_hertil/Abelsk_gruppe" title="Liste med alle sider som henviser hertil [j]" accesskey="j"><span>Hvad henviser hertil</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Speciel:Relaterede_%C3%A6ndringer/Abelsk_gruppe" rel="nofollow" title="Seneste ændringer af sider som denne side henviser til [k]" accesskey="k"><span>Beslægtede ændringer</span></a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=da&campaign=dk" title="Upload filer [u]" accesskey="u"><span>Upload fil</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Speciel:Specialsider" title="Liste over alle specialsider [q]" accesskey="q"><span>Specialsider</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&oldid=8178446" title="Permanent link til denne version af denne side"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&action=info" title="Yderligere oplysninger om denne side"><span>Sideinformation</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Speciel:Citer&page=Abelsk_gruppe&id=8178446&wpFormIdentifier=titleform" title="Information om, hvordan man kan citere denne side"><span>Referer til denne side</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speciel:UrlQ%C4%B1sald%C4%B1c%C4%B1s%C4%B1&url=https%3A%2F%2Fda.wikipedia.org%2Fwiki%2FAbelsk_gruppe"><span>Hent forkortet 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class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=Speciel:Bog&bookcmd=book_creator&referer=Abelsk+gruppe"><span>Lav en bog</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Speciel:DownloadAsPdf&page=Abelsk_gruppe&action=show-download-screen"><span>Download som PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Abelsk_gruppe&printable=yes" title="Printervenlig udgave af denne side [p]" accesskey="p"><span>Udskriftsvenlig udgave</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> I andre projekter </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Abelian_groups" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q181296" title="Link til tilknyttet emne i Wikidata [g]" accesskey="g"><span>Wikidata-element</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="Sideværktøjer"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Udseende"> <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">Udseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flyt til sidebjælken</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</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">Fra Wikipedia, den frie encyklopædi</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="da" dir="ltr"><p>En <b>abelsk gruppe</b> (eller en <b>kommutativ gruppe</b>) er inden for <a href="/wiki/Matematik" title="Matematik">matematikken</a> en <a href="/wiki/Gruppe_(matematik)" title="Gruppe (matematik)">gruppe</a>, (<i>G</i>, *), hvor den tilhørende <a href="/wiki/Operator" class="mw-redirect" title="Operator">operator</a>, *, er <a href="/wiki/Kommutativ" class="mw-redirect" title="Kommutativ">kommutativ</a>; for alle <i>a</i> og <i>b</i> i <i>G</i> skal gælde <i>a</i> * <i>b</i> = <i>b</i> * <i>a</i>. Grupperne er opkaldt efter den <a href="/wiki/Norge" title="Norge">norske</a> matematiker <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>. Teorien om abelske grupper er generelt simplere end for de ikke-abelske. Dog danner uendelige abelske grupper grundlag for et aktivt forskningsområde i matematikken. </p> <div class="mw-heading mw-heading2"><h2 id="Eksempler">Eksempler</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Abelsk_gruppe&veaction=edit&section=1" title="Redigér afsnit: Eksempler" class="mw-editsection-visualeditor"><span>redigér</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Abelsk_gruppe&action=edit&section=1" title="Edit section's source code: Eksempler"><span>rediger kildetekst</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Heltal" title="Heltal">Heltallene</a>, <b>Z</b>, danner en abelsk gruppe under addition (<i>n</i> + <i>m</i> = <i>m</i> + <i>n</i> for alle heltal <i>n</i> og <i>m</i>) og det samme gør mængden af <a href="/wiki/Modul%C3%A6r_aritmetik" title="Modulær aritmetik">heltal modulo <i>n</i></a>, <b>Z</b>/<i>n</i><b>Z</b>. Faktisk er enhver <a href="/w/index.php?title=Cyklisk_gruppe&action=edit&redlink=1" class="new" title="Cyklisk gruppe (ikke skrevet endnu)">cyklisk gruppe</a>, <i>G</i>, abelsk, da der for <i>x</i> og <i>y</i> i <i>G</i> gælder <i>xy</i> = <i>a</i><sup><i>m</i></sup><i>a</i><sup><i>n</i></sup> = <i>a</i><sup><i>m</i> + <i>n</i></sup> = <i>a</i><sup><i>n</i> + <i>m</i></sup> = <i>a</i><sup><i>n</i></sup><i>a</i><sup><i>m</i></sup> = <i>yx</i>. </p><p>Enhver <a href="/wiki/Ring_(matematik)" title="Ring (matematik)">ring</a> er en abelsk gruppe mht. additionsoperatoren. I en <a href="/wiki/Kommutativ_ring" title="Kommutativ ring">kommutativ ring</a> danner mængden af invertible elementer (dvs. <a href="/w/index.php?title=Enhed_(ringteori)&action=edit&redlink=1" class="new" title="Enhed (ringteori) (ikke skrevet endnu)">enhederne</a>) en abelsk <a href="/w/index.php?title=Multiplikativ_gruppe&action=edit&redlink=1" class="new" title="Multiplikativ gruppe (ikke skrevet endnu)">multiplikativ gruppe</a>. Specielt er de <a href="/wiki/Reelle_tal" title="Reelle tal">reelle tal</a> en abelsk gruppe under addition, mens de reelle tal fraregnet nul er en abelsk gruppe under multiplikation. </p><p>Enhver <a href="/wiki/Undergruppe" title="Undergruppe">undergruppe</a> af en abelsk gruppe er <a href="/w/index.php?title=Normal_undergruppe&action=edit&redlink=1" class="new" title="Normal undergruppe (ikke skrevet endnu)">normal</a>, så enhver undergruppe giver anledning til en <a href="/w/index.php?title=Kvotientgruppe&action=edit&redlink=1" class="new" title="Kvotientgruppe (ikke skrevet endnu)">kvotientgruppe</a>. Undergrupper, kvotientgrupper, og den <a href="/w/index.php?title=Direkte_sum_af_grupper&action=edit&redlink=1" class="new" title="Direkte sum af grupper (ikke skrevet endnu)">direkte sum</a> af abelske grupper er igen en abelsk gruppe. </p><p>Mængden af kvadratiske <a href="/wiki/Matrix" title="Matrix">matricer</a> danner en gruppe under multiplikation, der ikke er en abelsk gruppe, da matrixmultiplikation generelt ikke er kommutativt. </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r11759279">.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 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