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href="/wiki/%E7%89%B9%E6%AE%8A:%E5%90%BE%E6%9B%B8%E6%88%BF" title="此IP所修之議[n]" accesskey="n"><span>議</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="站台"> <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="目次" 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">目次</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至側邊欄</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隱藏</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">序言</div> </a> </li> <li id="toc-例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例"> <div class="vector-toc-text"> <span class="vector-toc-numb">一</span> <span>例</span> </div> </a> <ul id="toc-例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-有限生成交換群基本定理" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#有限生成交換群基本定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">二</span> <span>有限生成交換群基本定理</span> </div> </a> <ul id="toc-有限生成交換群基本定理-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="目次" 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="切換目次" > <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">切換目次</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">交換群</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="前往別種語言的條目。有五六種語言可用" > <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">五六種語言</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 – 南非荷蘭文" lang="af" hreflang="af" data-title="Abelse groep" data-language-autonym="Afrikaans" data-language-local-name="南非荷蘭文" 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 – 阿拉伯文" lang="ar" hreflang="ar" data-title="زمرة آبلية" data-language-autonym="العربية" data-language-local-name="阿拉伯文" 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 – 阿斯圖里亞文" lang="ast" hreflang="ast" data-title="Grupu abelianu" data-language-autonym="Asturianu" data-language-local-name="阿斯圖里亞文" 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 – 亞塞拜然文" lang="az" hreflang="az" data-title="Abel ortalamaları" data-language-autonym="Azərbaycanca" data-language-local-name="亞塞拜然文" 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 – 巴什喀爾文" lang="ba" hreflang="ba" data-title="Абель төркөмө" data-language-autonym="Башҡортса" data-language-local-name="巴什喀爾文" 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 – 白俄羅斯文" lang="be" hreflang="be" data-title="Абелева група" data-language-autonym="Беларуская" data-language-local-name="白俄羅斯文" 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 – 保加利亞文" lang="bg" hreflang="bg" data-title="Абелова група" data-language-autonym="Български" data-language-local-name="保加利亞文" 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 – 孟加拉文" lang="bn" hreflang="bn" data-title="আবেলীয় গ্রুপ" data-language-autonym="বাংলা" data-language-local-name="孟加拉文" 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 – 加泰蘭文" lang="ca" hreflang="ca" data-title="Grup abelià" data-language-autonym="Català" data-language-local-name="加泰蘭文" 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 – 中庫德文" lang="ckb" hreflang="ckb" data-title="گرووپی ئاڵوگۆڕی" data-language-autonym="کوردی" data-language-local-name="中庫德文" 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 – 捷克文" lang="cs" hreflang="cs" data-title="Abelova grupa" data-language-autonym="Čeština" data-language-local-name="捷克文" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Abelsk_gruppe" title="Abelsk gruppe – 丹麥文" lang="da" hreflang="da" data-title="Abelsk gruppe" data-language-autonym="Dansk" data-language-local-name="丹麥文" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Abelsche_Gruppe" title="Abelsche Gruppe – 德文" lang="de" hreflang="de" data-title="Abelsche Gruppe" data-language-autonym="Deutsch" data-language-local-name="德文" 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 – 希臘文" lang="el" hreflang="el" data-title="Αβελιανή ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="希臘文" 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 – 英文" lang="en" hreflang="en" data-title="Abelian group" data-language-autonym="English" data-language-local-name="英文" 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 – 世界文" lang="eo" hreflang="eo" data-title="Komuta grupo" data-language-autonym="Esperanto" data-language-local-name="世界文" 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 – 西班牙文" lang="es" hreflang="es" data-title="Grupo abeliano" data-language-autonym="Español" data-language-local-name="西班牙文" 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 – 愛沙尼亞文" lang="et" hreflang="et" data-title="Abeli rühm" data-language-autonym="Eesti" data-language-local-name="愛沙尼亞文" 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 – 巴斯克文" lang="eu" hreflang="eu" data-title="Talde abeldar" data-language-autonym="Euskara" data-language-local-name="巴斯克文" 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 – 波斯文" lang="fa" hreflang="fa" data-title="گروه آبلی" data-language-autonym="فارسی" data-language-local-name="波斯文" 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 – 芬蘭文" lang="fi" hreflang="fi" data-title="Abelin ryhmä" data-language-autonym="Suomi" data-language-local-name="芬蘭文" 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 – 法文" lang="fr" hreflang="fr" data-title="Groupe abélien" data-language-autonym="Français" data-language-local-name="法文" 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 – 加利西亞文" lang="gl" hreflang="gl" data-title="Grupo abeliano" data-language-autonym="Galego" data-language-local-name="加利西亞文" 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 – 希伯來文" lang="he" hreflang="he" data-title="חבורה אבלית" data-language-autonym="עברית" data-language-local-name="希伯來文" 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 – 克羅埃西亞文" lang="hr" hreflang="hr" data-title="Abelova grupa" data-language-autonym="Hrvatski" data-language-local-name="克羅埃西亞文" 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 – 匈牙利文" lang="hu" hreflang="hu" data-title="Abel-csoport" data-language-autonym="Magyar" data-language-local-name="匈牙利文" 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 – 亞美尼亞文" lang="hy" hreflang="hy" data-title="Աբելյան խումբ" data-language-autonym="Հայերեն" data-language-local-name="亞美尼亞文" 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 – 國際文" lang="ia" hreflang="ia" data-title="Gruppo abelian" data-language-autonym="Interlingua" data-language-local-name="國際文" 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 – 印尼文" lang="id" hreflang="id" data-title="Grup Abelian" data-language-autonym="Bahasa Indonesia" data-language-local-name="印尼文" 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 – 義大利文" lang="it" hreflang="it" data-title="Gruppo abeliano" data-language-autonym="Italiano" data-language-local-name="義大利文" 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 – 日文" lang="ja" hreflang="ja" data-title="アーベル群" data-language-autonym="日本語" data-language-local-name="日文" 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 – 卡拉卡爾帕克文" lang="kaa" hreflang="kaa" data-title="Abel toparı" data-language-autonym="Qaraqalpaqsha" data-language-local-name="卡拉卡爾帕克文" 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 – 韓文" lang="ko" hreflang="ko" data-title="아벨 군" data-language-autonym="한국어" data-language-local-name="韓文" 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 – 拉丁文" lang="la" hreflang="la" data-title="Caterva Abeliana" data-language-autonym="Latina" data-language-local-name="拉丁文" 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 – 馬來亞拉姆文" lang="ml" hreflang="ml" data-title="ക്രമഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="馬來亞拉姆文" 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 – 馬來文" lang="ms" hreflang="ms" data-title="Kumpulan Abel" data-language-autonym="Bahasa Melayu" data-language-local-name="馬來文" 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 – 荷蘭文" lang="nl" hreflang="nl" data-title="Abelse groep" data-language-autonym="Nederlands" data-language-local-name="荷蘭文" 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 – 耐諾斯克挪威文" lang="nn" hreflang="nn" data-title="Abelsk gruppe" data-language-autonym="Norsk nynorsk" data-language-local-name="耐諾斯克挪威文" 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 – 巴克摩挪威文" lang="nb" hreflang="nb" data-title="Abelsk gruppe" data-language-autonym="Norsk bokmål" data-language-local-name="巴克摩挪威文" 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 – 諾維亞文" lang="nov" hreflang="nov" data-title="Abelan grupe" data-language-autonym="Novial" data-language-local-name="諾維亞文" 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 – 波蘭文" lang="pl" hreflang="pl" data-title="Grupa przemienna" data-language-autonym="Polski" data-language-local-name="波蘭文" 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 – 葡萄牙文" lang="pt" hreflang="pt" data-title="Grupo abeliano" data-language-autonym="Português" data-language-local-name="葡萄牙文" 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 – 羅馬尼亞文" lang="ro" hreflang="ro" data-title="Grup abelian" data-language-autonym="Română" data-language-local-name="羅馬尼亞文" 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 – 俄文" lang="ru" hreflang="ru" data-title="Абелева группа" data-language-autonym="Русский" data-language-local-name="俄文" 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 – 塞爾維亞克羅埃西亞文" lang="sh" hreflang="sh" data-title="Abelova grupa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="塞爾維亞克羅埃西亞文" 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 – 斯洛伐克文" lang="sk" hreflang="sk" data-title="Abelovská grupa" data-language-autonym="Slovenčina" data-language-local-name="斯洛伐克文" 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 – 斯洛維尼亞文" lang="sl" hreflang="sl" data-title="Abelova grupa" data-language-autonym="Slovenščina" data-language-local-name="斯洛維尼亞文" 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 – 塞爾維亞文" lang="sr" hreflang="sr" data-title="Абелова група" data-language-autonym="Српски / srpski" data-language-local-name="塞爾維亞文" 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 – 瑞典文" lang="sv" hreflang="sv" data-title="Abelsk grupp" data-language-autonym="Svenska" data-language-local-name="瑞典文" 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 – 坦米爾文" lang="ta" hreflang="ta" data-title="பரிமாற்றுக் குலம்" data-language-autonym="தமிழ்" data-language-local-name="坦米爾文" 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 – 土耳其文" lang="tr" hreflang="tr" data-title="Abel grubu" data-language-autonym="Türkçe" data-language-local-name="土耳其文" 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 – 烏克蘭文" lang="uk" hreflang="uk" data-title="Абелева група" data-language-autonym="Українська" data-language-local-name="烏克蘭文" 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 – 越南文" lang="vi" hreflang="vi" data-title="Nhóm giao hoán" data-language-autonym="Tiếng Việt" data-language-local-name="越南文" 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 – 中文" lang="zh" hreflang="zh" data-title="阿贝尔群" data-language-autonym="中文" data-language-local-name="中文" class="interlanguage-link-target"><span>中文</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 – 粵語" lang="yue" hreflang="yue" data-title="阿貝爾羣" data-language-autonym="粵語" data-language-local-name="粵語" 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 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data-event-name="pinnable-header.vector-page-tools.unpin">隱藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多選項" > <div class="vector-menu-heading"> 操作 </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/%E4%BA%A4%E6%8F%9B%E7%BE%A4"><span>閱</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%BA%A4%E6%8F%9B%E7%BE%A4&action=edit" title="編輯此頁面的原始碼[e]" accesskey="e"><span>纂</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E4%BA%A4%E6%8F%9B%E7%BE%A4&action=history"><span>覽史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 一般 </div> <div class="vector-menu-content"> 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class="cdx-dialog__header__title">頁面版本狀態</h2><p class="cdx-dialog__header__subtitle">該頁面尚未檢查</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
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							cdx-fr-css-icon--close"></span></button></header><div class="cdx-dialog__body">此為底本，未經<a href="/wiki/%E8%AA%AA%E6%98%8E:%E6%A0%A1%E5%B0%8D" title="說明:校對">審校</a>。</div></div><div tabindex="0"></div></div><div class="mw-content-ltr mw-parser-output" lang="lzh" dir="ltr"><p><b>交換群</b>者，<a href="/wiki/%E4%BA%8C%E5%85%83%E9%81%8B%E7%AE%97" title="二元運算">乘法</a>合<a href="/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" class="mw-redirect" title="交換律">交換律</a>之<a href="/wiki/%E7%BE%A4_(%E4%BB%A3%E6%95%B8)" title="群 (代數)">群</a>也，亦曰<b>阿貝爾群</b>。可以加法謂之，則其單位元曰「零」，逆改曰負。 </p><p>初，<a href="/w/index.php?title=%E9%98%BF%E8%B2%9D%E7%88%BE&action=edit&redlink=1" class="new" title="阿貝爾（查無此頁）">阿貝爾</a>以<a href="/w/index.php?title=%E7%BD%AE%E6%8F%9B&action=edit&redlink=1" class="new" title="置換（查無此頁）">置換</a>之不可<a href="/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" class="mw-redirect" title="交換律">交換</a>，證五次<a href="/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式">多項式</a>無簡約解（<a href="/w/index.php?title=%E9%98%BF%E8%B2%9D%E7%88%BE%E2%80%93%E9%AD%AF%E8%8A%AC%E5%B0%BC%E5%AE%9A%E7%90%86&action=edit&redlink=1" class="new" title="阿貝爾–魯芬尼定理（查無此頁）">阿貝爾–魯芬尼定理</a>）。然天妒英才，終年不過三十。泰西疇人念其功績，名交換群為阿貝爾群也。 </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="例"><span id=".E4.BE.8B"></span>例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%BA%A4%E6%8F%9B%E7%BE%A4&action=edit&section=1" title="所纂章名：例"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%E6%95%B4%E6%95%B8" title="整數">整數</a>合加法，<a href="/wiki/%E5%88%86%E6%95%B8" title="分數">分數</a>合加法，<a href="/wiki/%E6%AD%A3%E6%95%B8" title="正數">正數</a>合乘法，非零<a href="/wiki/%E8%A4%87%E6%95%B8" title="複數">複數</a>合乘法，<a href="/w/index.php?title=%E5%90%8C%E9%A4%98&action=edit&redlink=1" class="new" title="同餘（查無此頁）">同餘</a>集合加法，俱交換群也。 </p><p><br /> </p> <div class="mw-heading mw-heading2"><h2 id="有限生成交換群基本定理"><span id=".E6.9C.89.E9.99.90.E7.94.9F.E6.88.90.E4.BA.A4.E6.8F.9B.E7.BE.A4.E5.9F.BA.E6.9C.AC.E5.AE.9A.E7.90.86"></span>有限生成交換群基本定理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%BA%A4%E6%8F%9B%E7%BE%A4&action=edit&section=2" title="所纂章名：有限生成交換群基本定理"><span>纂</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>取一集合，含此集之群最小者，曰集之<a href="/wiki/%E7%94%9F%E6%88%90" title="生成">生成</a>群。<a href="/w/index.php?title=%E6%9C%89%E9%99%90%E9%9B%86&action=edit&redlink=1" class="new" title="有限集（查無此頁）">有限集</a>之生成群，曰有限生成群。 </p><p>「有限生成交換群基本定理」云：有限生成交換群，乃整數群與<a href="/w/index.php?title=%E5%90%8C%E9%A4%98&action=edit&redlink=1" class="new" title="同餘（查無此頁）">同餘</a>群之直積也。（「<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\cong \mathbb {Z} \oplus \cdots \oplus \mathbb {Z} \oplus \mathbb {Z} _{n_{1}}\oplus \cdots \oplus \mathbb {Z} _{n_{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo>≅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G\cong \mathbb {Z} \oplus \cdots \oplus \mathbb {Z} \oplus \mathbb {Z} _{n_{1}}\oplus \cdots \oplus \mathbb {Z} _{n_{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/314bd9a7811f24a75f1571008a1a1bf34f590bf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.81ex; height:2.843ex;" alt="{\displaystyle G\cong \mathbb {Z} \oplus \cdots \oplus \mathbb {Z} \oplus \mathbb {Z} _{n_{1}}\oplus \cdots \oplus \mathbb {Z} _{n_{r}}}"></span>」） </p><p>因而得證，有限交換群乃同餘群之直積也。 </p></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">取自「<a dir="ltr" href="https://zh-classical.wikipedia.org/wiki/交換群">https://zh-classical.wikipedia.org/wiki/交換群</a>」</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%E7%89%B9%E6%AE%8A:%E9%A1%9E" title="特殊:類">一類</a>：<ul><li><a href="/wiki/%E5%88%86%E9%A1%9E:%E4%BB%A3%E6%95%B8" title="分類:代數">代數</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-copyright">凡我維基之文，悉為<a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">共享創意授權</a>。翻印增刪，皆須引據，並同道為之。然條款繁多，不一而足。請閱<a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">全文</a>，以得其詳。<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">隱私通例</a></li> <li id="footer-places-about"><a href="/wiki/%E6%96%87%E8%A8%80%E7%B6%AD%E5%9F%BA%E5%A4%A7%E5%85%B8">大典自序</a></li> <li id="footer-places-disclaimers"><a href="https://en.wikipedia.org/wiki/Wikipedia:General_disclaimer">免責告示</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">行為準則</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">司空津逮</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/zh-classical.wikipedia.org">彙統</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">鑑識聲明</a></li> <li id="footer-places-mobileview"><a href="//zh-classical.m.wikipedia.org/w/index.php?title=%E4%BA%A4%E6%8F%9B%E7%BE%A4&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">掌中書卷</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7fc47fc68d-xm24l","wgBackendResponseTime":173,"wgPageParseReport":{"limitreport":{"cputime":"0.010","walltime":"0.044","ppvisitednodes":{"value":22,"limit":1000000},"postexpandincludesize":{"value":0,"limit":2097152},"templateargumentsize":{"value":0,"limit":2097152},"expansiondepth":{"value":2,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":36,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 0.000 1 -total"]},"cachereport":{"origin":"mw-api-int.codfw.main-648bd44df8-z86qh","timestamp":"20241115192144","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u4ea4\u63db\u7fa4","url":"https:\/\/zh-classical.wikipedia.org\/wiki\/%E4%BA%A4%E6%8F%9B%E7%BE%A4","sameAs":"http:\/\/www.wikidata.org\/entity\/Q181296","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q181296","author":{"@type":"Organization","name":"\u7dad\u57fa\u5a92\u9ad4\u5c08\u6848\u8ca2\u737b\u8005"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2007-10-28T23:42:42Z"}</script> </body> </html>