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href="/w/index.php?title=%E7%89%B9%E5%88%A5:%E3%83%AD%E3%82%B0%E3%82%A4%E3%83%B3&returnto=%E8%87%AA%E6%98%8E%E7%BE%A4" title="ログインすることを推奨します。ただし、必須ではありません。 [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>ログイン</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"> ログアウトした編集者のページ <a href="/wiki/Wikipedia:%E3%82%A6%E3%82%A3%E3%82%AD%E3%83%9A%E3%83%87%E3%82%A3%E3%82%A2%E3%81%B8%E3%82%88%E3%81%86%E3%81%93%E3%81%9D" aria-label="編集の詳細"><span>もっと詳しく</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/%E7%89%B9%E5%88%A5:%E8%87%AA%E5%88%86%E3%81%AE%E6%8A%95%E7%A8%BF%E8%A8%98%E9%8C%B2" title="このIPアドレスからなされた編集の一覧 [y]" accesskey="y"><span>投稿記録</span></a></li><li id="pt-anontalk" class="mw-list-item"><a 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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">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">非表示</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">ページ先頭</div> </a> </li> <li id="toc-性質" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性質"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>性質</span> </div> </a> <ul id="toc-性質-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-関連項目" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#関連項目"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>関連項目</span> </div> </a> <ul id="toc-関連項目-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-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="目次の表示・非表示を切り替え" > <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">目次の表示・非表示を切り替え</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">自明群</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="特定の記事の別の言語版に移動します。利用可能な言語23件" > <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-23" 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">23の言語版</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%AF%D9%8A%D9%87%D9%8A%D8%A9" title="アラビア語: زمرة بديهية" lang="ar" hreflang="ar" data-title="زمرة بديهية" data-language-autonym="العربية" data-language-local-name="アラビア語" 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_trivial" title="カタロニア語: Grup trivial" lang="ca" hreflang="ca" data-title="Grup trivial" data-language-autonym="Català" data-language-local-name="カタロニア語" 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/Trivi%C3%A1ln%C3%AD_grupa" title="チェコ語: Triviální grupa" lang="cs" hreflang="cs" data-title="Triviální grupa" data-language-autonym="Čeština" data-language-local-name="チェコ語" 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/Triviel_gruppe" title="デンマーク語: Triviel gruppe" lang="da" hreflang="da" data-title="Triviel gruppe" data-language-autonym="Dansk" data-language-local-name="デンマーク語" 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/Triviale_Gruppe" title="ドイツ語: Triviale Gruppe" lang="de" hreflang="de" data-title="Triviale Gruppe" data-language-autonym="Deutsch" data-language-local-name="ドイツ語" 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%A4%CE%B5%CF%84%CF%81%CE%B9%CE%BC%CE%BC%CE%AD%CE%BD%CE%B7_%CE%BF%CE%BC%CE%AC%CE%B4%CE%B1" title="ギリシャ語: Τετριμμένη ομάδα" lang="el" hreflang="el" data-title="Τετριμμένη ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="ギリシャ語" 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/Trivial_group" title="英語: Trivial group" lang="en" hreflang="en" data-title="Trivial group" data-language-autonym="English" data-language-local-name="英語" 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_trivial" title="スペイン語: Grupo trivial" lang="es" hreflang="es" data-title="Grupo trivial" data-language-autonym="Español" data-language-local-name="スペイン語" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Triviaali_ryhm%C3%A4" title="フィンランド語: Triviaali ryhmä" lang="fi" hreflang="fi" data-title="Triviaali ryhmä" data-language-autonym="Suomi" data-language-local-name="フィンランド語" 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_trivial" title="フランス語: Groupe trivial" lang="fr" hreflang="fr" data-title="Groupe trivial" data-language-autonym="Français" data-language-local-name="フランス語" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Trivi%C3%A1lis_csoport" title="ハンガリー語: Triviális csoport" lang="hu" hreflang="hu" data-title="Triviális csoport" data-language-autonym="Magyar" data-language-local-name="ハンガリー語" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EB%AA%85%EA%B5%B0" title="韓国語: 자명군" lang="ko" hreflang="ko" data-title="자명군" data-language-autonym="한국어" data-language-local-name="韓国語" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Triviale_groep" title="オランダ語: Triviale groep" lang="nl" hreflang="nl" data-title="Triviale groep" data-language-autonym="Nederlands" data-language-local-name="オランダ語" 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_trywialna" title="ポーランド語: Grupa trywialna" lang="pl" hreflang="pl" data-title="Grupa trywialna" data-language-autonym="Polski" data-language-local-name="ポーランド語" 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_trivial" title="ポルトガル語: Grupo trivial" lang="pt" hreflang="pt" data-title="Grupo trivial" data-language-autonym="Português" data-language-local-name="ポルトガル語" 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_trivial" title="ルーマニア語: Grup trivial" lang="ro" hreflang="ro" data-title="Grup trivial" data-language-autonym="Română" data-language-local-name="ルーマニア語" 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%A2%D1%80%D0%B8%D0%B2%D0%B8%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="ロシア語: Тривиальная группа" lang="ru" hreflang="ru" data-title="Тривиальная группа" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Trivialna_grupa" title="スロベニア語: Trivialna grupa" lang="sl" hreflang="sl" data-title="Trivialna grupa" data-language-autonym="Slovenščina" data-language-local-name="スロベニア語" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Trivialgrupp" title="スウェーデン語: Trivialgrupp" lang="sv" hreflang="sv" data-title="Trivialgrupp" data-language-autonym="Svenska" data-language-local-name="スウェーデン語" 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%8E%E0%AE%B3%E0%AE%BF%E0%AE%AF_%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="タミル語: எளிய குலம்" lang="ta" hreflang="ta" data-title="எளிய குலம்" data-language-autonym="தமிழ்" data-language-local-name="タミル語" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A2%D1%80%D0%B8%D0%B2%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="ウクライナ語: Тривіальна група" lang="uk" hreflang="uk" data-title="Тривіальна група" data-language-autonym="Українська" data-language-local-name="ウクライナ語" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%95%B6%E7%84%B6%E7%BE%A4" title="中国語: 當然群" lang="zh" hreflang="zh" data-title="當然群" data-language-autonym="中文" data-language-local-name="中国語" 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/%E5%B9%B3%E5%87%A1%E7%BE%A3" title="広東語: 平凡羣" lang="yue" hreflang="yue" data-title="平凡羣" data-language-autonym="粵語" data-language-local-name="広東語" 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 href="https://www.wikidata.org/wiki/Special:EntityPage/Q568687#sitelinks-wikipedia" title="言語間リンクを編集" class="wbc-editpage">リンクを編集</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="名前空間"> <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/%E8%87%AA%E6%98%8E%E7%BE%A4" title="本文を閲覧 [c]" accesskey="c"><span>ページ</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/%E3%83%8E%E3%83%BC%E3%83%88:%E8%87%AA%E6%98%8E%E7%BE%A4" rel="discussion" title="本文ページについての議論 [t]" accesskey="t"><span>ノート</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="別の言語に切り替える" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">日本語</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="表示"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E8%87%AA%E6%98%8E%E7%BE%A4"><span>閲覧</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=edit" title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=history" title="このページの過去の版 [h]" accesskey="h"><span>履歴表示</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="ページツール"> <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="ツール" > <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">ツール</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">ツール</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">非表示</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="その他の操作" > <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/%E8%87%AA%E6%98%8E%E7%BE%A4"><span>閲覧</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=edit" title="このページのソースコードを編集する [e]" accesskey="e"><span>編集</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=history"><span>履歴表示</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 全般 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%AA%E3%83%B3%E3%82%AF%E5%85%83/%E8%87%AA%E6%98%8E%E7%BE%A4" title="ここにリンクしている全ウィキページの一覧 [j]" accesskey="j"><span>リンク元</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a 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href="https://www.wikidata.org/wiki/Special:EntityPage/Q568687" title="関連付けられたデータリポジトリ項目へのリンク [g]" accesskey="g"><span>ウィキデータ項目</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="ページツール"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="表示"> <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">表示</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">サイドバーに移動</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">非表示</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">出典: フリー百科事典『ウィキペディア（Wikipedia）』</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="ja" dir="ltr"><p><a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a>において、<b>自明群</b>、<b>自明な群</b> (trivial group)、<b>単位群</b> はただ1つの元からなる<a href="/wiki/%E7%BE%A4_(%E6%95%B0%E5%AD%A6)" title="群 (数学)">群</a>である。すべてのそのような群は<a href="/wiki/%E5%90%8C%E5%9E%8B" class="mw-redirect" title="同型">同型</a>であるので、英語などではしばしば定冠詞をつけて <i>the</i> trivial group などと呼ばれる。自明群のただ1つの元は<a href="/wiki/%E5%8D%98%E4%BD%8D%E5%85%83" title="単位元">単位元</a>であるので普通 0, 1, <i>e</i> のように文脈に応じて表記される。群の演算が ∗ であれば <span lang="en" class="texhtml"><i>e</i> ∗ <i>e</i> = <i>e</i></span> によって定義される。 </p><p>同様に定義される<b>自明モノイド</b> (trivial monoid) もまた群である。その唯一の元がそれ自身の逆元でありしたがって自明群と同じであるからである。 </p><p>自明群を<a href="/wiki/%E7%A9%BA%E9%9B%86%E5%90%88" title="空集合">空集合</a>と混同してはならない。（これは元を全くもたず、単位元を欠くため、群にはなりえない。） </p><p>任意の群 <i>G</i> が与えられると、単位元のみからなる部分集合は、それ自身が自明群である <i>G</i> の<a href="/wiki/%E9%83%A8%E5%88%86%E7%BE%A4" title="部分群">部分群</a>であり、<i>G</i> の<b>自明な部分群</b> (trivial subgroup) と呼ばれる。また、<i>G</i> 自身も明らかに <i>G</i> の部分群であるので、<i>G</i> も自明な部分群と呼ばれることがあるが、これは著者によって異なるので注意が必要である。群によってはこれら以外にも自明に部分群になるものがあるが、それらは自明な部分群とは呼ばれない。 </p><p>"<i>G</i> は非自明な真の部分群をもたない" (<i>G</i> has no nontrivial proper subgroups) という言い回しが意味するのは、<i>G</i> のすべての部分群は自明群 {<i>e</i>} および群 <i>G</i> 自身であるということである。 </p> <div class="mw-heading mw-heading2"><h2 id="性質"><span id=".E6.80.A7.E8.B3.AA"></span>性質</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=edit&section=1" title="節を編集: 性質"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>自明群は<a href="/wiki/%E4%BD%8D%E6%95%B0_(%E7%BE%A4%E8%AB%96)" title="位数 (群論)">位数</a> 1 の<a href="/wiki/%E5%B7%A1%E5%9B%9E%E7%BE%A4" title="巡回群">巡回群</a>である。そのようなものとして自明群は Z<sub>1</sub> あるいは C<sub>1</sub> と書ける。群の演算が和であれば、自明群は通常単に 0 と表記される。群演算が積であれば 1 が自明群の表記となる。 </p><p>群の自明な部分群（および群自身）は<a href="/wiki/%E6%AD%A3%E8%A6%8F%E9%83%A8%E5%88%86%E7%BE%A4" title="正規部分群">正規部分群</a>である。正規部分群が自明な部分群と群自身しかないような非自明な群は<a href="/wiki/%E5%8D%98%E7%B4%94%E7%BE%A4" title="単純群">単純群</a>と呼ばれる。 </p><p>自明群は<a href="/wiki/%E7%BE%A4%E3%81%AE%E5%9C%8F" title="群の圏">群の圏</a>において<a href="/wiki/%E9%9B%B6%E5%AF%BE%E8%B1%A1" class="mw-redirect" title="零対象">零対象</a>として振る舞う。つまり、それは<a href="/wiki/%E5%A7%8B%E5%AF%BE%E8%B1%A1" class="mw-redirect" title="始対象">始対象</a>かつ<a href="/wiki/%E7%B5%82%E5%AF%BE%E8%B1%A1" class="mw-redirect" title="終対象">終対象</a>である。 </p> <div class="mw-heading mw-heading2"><h2 id="関連項目"><span id=".E9.96.A2.E9.80.A3.E9.A0.85.E7.9B.AE"></span>関連項目</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=edit&section=2" title="節を編集: 関連項目"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span title="リンク先の項目はまだありません。新規の執筆や他言語版からの翻訳が望まれます。"><a href="/w/index.php?title=%E5%B0%8F%E3%81%95%E3%81%84%E7%BE%A4%E3%81%AE%E4%B8%80%E8%A6%A7&action=edit&redlink=1" class="new" title="「小さい群の一覧」 (存在しないページ)">小さい群の一覧</a><span style="font-size: 0.77em; font-weight: normal;" class="noprint">（<a href="https://en.wikipedia.org/wiki/List_of_small_groups" class="extiw" title="en:List of small groups">英語版</a>）</span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E8%87%AA%E6%98%8E%E7%BE%A4&action=edit&section=3" title="節を編集: 参考文献"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Trivial_Group"><style data-mw-deduplicate="TemplateStyles:r101121245">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation.cs-ja1 q,.mw-parser-output .citation.cs-ja2 q{quotes:"「""」""『""』"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:var(--color-success,#3a3);margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style><cite id="CITEREFRowland,_Todd_and_Weisstein,_Eric_W." class="citation web cs1 cs1-prop-foreign-lang-source">Rowland, Todd and Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TrivialGroup.html">"Trivial Group"</a>. <i>mathworld.wolfram.com</i> (英語).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=mathworld.wolfram.com&rft.atitle=Trivial+Group&rft.au=Rowland%2C+Todd+and+Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FTrivialGroup.html&rfr_id=info%3Asid%2Fja.wikipedia.org%3A%E8%87%AA%E6%98%8E%E7%BE%A4" class="Z3988"></span></span></li> <li><cite style="font-style:normal" class="citation book">森田康夫『代数概論』裳華房、1987年。<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="/wiki/ISBN" title="ISBN">ISBN</a> <a href="/wiki/%E7%89%B9%E5%88%A5:%E6%96%87%E7%8C%AE%E8%B3%87%E6%96%99/978-4-7853-1311-1" title="特別:文献資料/978-4-7853-1311-1">978-4-7853-1311-1</a>。</cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%E4%BB%A3%E6%95%B0%E6%A6%82%E8%AB%96&rft.aulast=%E6%A3%AE%E7%94%B0%E5%BA%B7%E5%A4%AB&rft.au=%E6%A3%AE%E7%94%B0%E5%BA%B7%E5%A4%AB&rft.date=1987&rft.pub=%E8%A3%B3%E8%8F%AF%E6%88%BF&rft.isbn=978-4-7853-1311-1&rfr_id=info:sid/ja.wikipedia.org:%E8%87%AA%E6%98%8E%E7%BE%A4"><span style="display: none;"> </span></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">「<a dir="ltr" href="https://ja.wikipedia.org/w/index.php?title=自明群&oldid=67757383">https://ja.wikipedia.org/w/index.php?title=自明群&oldid=67757383</a>」から取得</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a 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