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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 href="/wiki/%E7%89%B9%E5%88%A5:%E3%83%88%E3%83%BC%E3%82%AF%E3%83%9A%E3%83%BC%E3%82%B8" title="このIPアドレスからなされた編集についての議論 [n]" accesskey="n"><span>トーク</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="サイト"> <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">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-四元数群" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#四元数群"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>四元数群</span> </div> </a> <ul id="toc-四元数群-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ブラウアー・鈴木の定理" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ブラウアー・鈴木の定理"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>ブラウアー・鈴木の定理</span> </div> </a> <ul id="toc-ブラウアー・鈴木の定理-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-脚注" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#脚注"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>脚注</span> </div> </a> <ul id="toc-脚注-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">4</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部リンク" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部リンク"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>外部リンク</span> </div> </a> <ul id="toc-外部リンク-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">一般四元数群</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8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-8" 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">8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-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Dizyklische_Gruppe" title="ドイツ語: Dizyklische Gruppe" lang="de" hreflang="de" data-title="Dizyklische 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Dicyclic_group" title="英語: Dicyclic group" lang="en" hreflang="en" data-title="Dicyclic 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-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_dicyclique" title="フランス語: Groupe dicyclique" lang="fr" hreflang="fr" data-title="Groupe dicyclique" 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-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_disiklik" title="インドネシア語: Grup disiklik" lang="id" hreflang="id" data-title="Grup disiklik" data-language-autonym="Bahasa Indonesia" data-language-local-name="インドネシア語" 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_diciclico" title="イタリア語: Gruppo diciclico" lang="it" hreflang="it" data-title="Gruppo diciclico" data-language-autonym="Italiano" data-language-local-name="イタリア語" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8C%8D%EC%88%9C%ED%99%98%EA%B5%B0" title="韓国語: 쌍순환군" lang="ko" hreflang="ko" data-title="쌍순환군" 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/Dicyclische_groep" title="オランダ語: Dicyclische groep" lang="nl" hreflang="nl" data-title="Dicyclische 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%86%D0%B8%D0%BA%D0%BB%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D0%B0" title="ロシア語: Дициклическая группа" lang="ru" hreflang="ru" data-title="Дициклическая группа" data-language-autonym="Русский" data-language-local-name="ロシア語" class="interlanguage-link-target"><span>Русский</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/Q3777914#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/%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4" title="本文を閲覧 [c]" accesskey="c"><span>ページ</span></a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E3%83%8E%E3%83%BC%E3%83%88:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&action=edit&redlink=1" rel="discussion" class="new" title="「本文ページについての議論」 (存在しないページ) [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/%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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/%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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/%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%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 href="/wiki/%E7%89%B9%E5%88%A5:%E9%96%A2%E9%80%A3%E3%83%9A%E3%83%BC%E3%82%B8%E3%81%AE%E6%9B%B4%E6%96%B0%E7%8A%B6%E6%B3%81/%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4" rel="nofollow" title="このページからリンクしているページの最近の更新 [k]" accesskey="k"><span>関連ページの更新状況</span></a></li><li id="t-upload" class="mw-list-item"><a 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</div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q3777914" 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" 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href="/w/index.php?title=%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&redirect=no" class="mw-redirect" title="四元数群">四元数群</a>から転送）</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ja" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r101346560">.mw-parser-output .hatnote{margin:0.5em 0;padding:3px 2em;background-color:transparent;border-bottom:1px solid #a2a9b1;font-size:90%}html.skin-theme-clientpref-night .mw-parser-output .hatnote>table{color:inherit}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .hatnote>table{color:inherit}}</style><div class="hatnote dablink noprint"><table style="width:100%; background:transparent;"> <tbody><tr><td style="width:25px;"><span typeof="mw:File"><a href="/wiki/%E3%83%95%E3%82%A1%E3%82%A4%E3%83%AB:Confusion_grey.svg" class="mw-file-description" title="曖昧さ回避"><img alt="曖昧さ回避" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/25px-Confusion_grey.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/38px-Confusion_grey.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/Confusion_grey.svg/50px-Confusion_grey.svg.png 2x" data-file-width="260" data-file-height="200" /></a></span></td> <td><a href="/wiki/%E3%82%AF%E3%83%A9%E3%82%A4%E3%83%B3%E3%81%AE%E5%9B%9B%E5%85%83%E7%BE%A4" title="クラインの四元群">クラインの四元群</a>（<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">four-group</span>, <a href="/wiki/%E3%83%89%E3%82%A4%E3%83%84%E8%AA%9E" title="ドイツ語">独</a>: <span lang="de">Vierergruppe</span>）とは異なります。</td> </tr></tbody></table></div> <p><a href="/wiki/%E6%95%B0%E5%AD%A6" title="数学">数学</a>において、<b>一般四元数群</b><sup id="cite_ref-FOOTNOTE森197563_1-0" class="reference"><a href="#cite_note-FOOTNOTE森197563-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTE岩波数学辞典20071530_2-0" class="reference"><a href="#cite_note-FOOTNOTE岩波数学辞典20071530-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>（いっぱんしげんすうぐん、<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">generalized quaternion group</span>）とは、<a href="#四元数群">四元数群</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{8}=\{\pm 1,\pm i,\pm j,\pm k\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mi>i</mi> <mo>,</mo> <mo>±</mo> <mi>j</mi> <mo>,</mo> <mo>±</mo> <mi>k</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{8}=\{\pm 1,\pm i,\pm j,\pm k\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3179e3218b1081ff19e7c745e6215e50201fb8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.785ex; height:2.843ex;" alt="{\displaystyle Q_{8}=\{\pm 1,\pm i,\pm j,\pm k\}}"></span></dd></dl> <p>を一般化した<a href="/wiki/%E6%9C%89%E9%99%90%E7%BE%A4" title="有限群">有限群</a>のこと。これは </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{4m}=\langle \,a,b\mid a^{2m}=1,\ b^{2}=a^{m},\ b^{-1}ab=a^{-1}\,\rangle \qquad (m>1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mspace width="thinmathspace" /> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∣</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mtext> </mtext> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>a</mi> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <mi>m</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{4m}=\langle \,a,b\mid a^{2m}=1,\ b^{2}=a^{m},\ b^{-1}ab=a^{-1}\,\rangle \qquad (m>1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7735d797d7b99a2936e61ede91641c8415735a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.462ex; height:3.176ex;" alt="{\displaystyle Q_{4m}=\langle \,a,b\mid a^{2m}=1,\ b^{2}=a^{m},\ b^{-1}ab=a^{-1}\,\rangle \qquad (m>1)}"></span></dd></dl> <p>という<a href="/wiki/%E7%BE%A4%E3%81%AE%E8%A1%A8%E7%A4%BA" title="群の表示">表示</a>で定義される<sup id="cite_ref-FOOTNOTEATLAS1985xx_4-0" class="reference"><a href="#cite_note-FOOTNOTEATLAS1985xx-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>、<a href="/wiki/%E4%BD%8D%E6%95%B0_(%E7%BE%A4%E8%AB%96)" title="位数 (群論)">位数</a> <span lang="en" class="texhtml">4<i>m</i></span> で、位数が <span lang="en" class="texhtml">2</span> である部分群（<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 Z(Q_{4m})=\langle a^{m}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>m</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(Q_{4m})=\langle a^{m}\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c67907ed62f837fd6f1e0a72583e42234abeaca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.638ex; height:2.843ex;" alt="{\displaystyle Z(Q_{4m})=\langle a^{m}\rangle }"></span>）を唯一つ持つ群である<sup id="cite_ref-FOOTNOTE近藤199131,_382_6-0" class="reference"><a href="#cite_note-FOOTNOTE近藤199131,_382-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>。（2群の場合しか考えないこともある<sup id="cite_ref-FOOTNOTE岩波数学辞典20071530_2-1" class="reference"><a href="#cite_note-FOOTNOTE岩波数学辞典20071530-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTE鈴木1977255_7-0" class="reference"><a href="#cite_note-FOOTNOTE鈴木1977255-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTE森197563_1-1" class="reference"><a href="#cite_note-FOOTNOTE森197563-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>。この場合、位数 <span lang="en" class="texhtml">2<sup><i>n</i></sup></span> の一般四元数群を <span lang="en" class="texhtml"><i>Q</i><sub><i>n</i></sub></span> と書く流儀もあり<sup id="cite_ref-FOOTNOTE森197563_1-2" class="reference"><a href="#cite_note-FOOTNOTE森197563-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>、注意が必要である。）<a href="/wiki/%E7%BE%A4%E3%81%AE%E7%94%9F%E6%88%90%E5%85%83" class="mw-redirect" title="群の生成元">群の生成元</a>を </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\mapsto {\begin{bmatrix}e^{\pi i/m}&0\\0&e^{-\pi i/m}\end{bmatrix}},\quad b\mapsto {\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>π</mi> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>m</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\mapsto {\begin{bmatrix}e^{\pi i/m}&0\\0&e^{-\pi i/m}\end{bmatrix}},\quad b\mapsto {\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5717f1279059b02702dda02e2166ce40240761a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.461ex; height:6.509ex;" alt="{\displaystyle a\mapsto {\begin{bmatrix}e^{\pi i/m}&0\\0&e^{-\pi i/m}\end{bmatrix}},\quad b\mapsto {\begin{bmatrix}0&1\\-1&0\end{bmatrix}}}"></span></dd></dl> <p>のように対応させることで、<a href="/wiki/%E5%BF%A0%E5%AE%9F%E8%A1%A8%E7%8F%BE" title="忠実表現">忠実な行列表現</a>を得ることができる。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="四元数群"><span id=".E5.9B.9B.E5.85.83.E6.95.B0.E7.BE.A4"></span>四元数群</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&action=edit&section=1" title="節を編集: 四元数群"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>四元数群</b>（しげんすうぐん、<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">quaternion group</span>）は </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{8}=\langle \,i,j,k\mid i^{2}=j^{2}=k^{2}=ijk\,\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>∣</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{8}=\langle \,i,j,k\mid i^{2}=j^{2}=k^{2}=ijk\,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4b4752bf7d19646b9c0756f5de956a6fcd60f78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.953ex; height:3.176ex;" alt="{\displaystyle Q_{8}=\langle \,i,j,k\mid i^{2}=j^{2}=k^{2}=ijk\,\rangle }"></span></dd></dl> <p>という<a href="/wiki/%E7%BE%A4%E3%81%AE%E8%A1%A8%E7%A4%BA" title="群の表示">表示</a>で定義される<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>。これは<a href="/wiki/%E4%BD%8D%E6%95%B0_(%E7%BE%A4%E8%AB%96)" title="位数 (群論)">位数</a> <span lang="en" class="texhtml">8</span> の非<a href="/wiki/%E5%8F%AF%E6%8F%9B%E7%BE%A4" class="mw-redirect" title="可換群">可換群</a>で、すべての真<a href="/wiki/%E9%83%A8%E5%88%86%E7%BE%A4" title="部分群">部分群</a>は<a href="/wiki/%E5%B7%A1%E5%9B%9E%E7%BE%A4" title="巡回群">巡回的</a>である。元 <span lang="en" class="texhtml"><i>ijk</i> ∈ <i>Q</i><sub>8</sub></span> は唯一つの<a href="/wiki/%E5%AF%BE%E5%90%88" title="対合">対合</a>で<a href="/wiki/%E7%BE%A4%E3%81%AE%E4%B8%AD%E5%BF%83" title="群の中心">中心的</a>であり、 <span lang="en" class="texhtml">−1</span> と書かれることも多い。これらの記号はハミルトンの<a href="/wiki/%E5%9B%9B%E5%85%83%E6%95%B0%E7%92%B0" title="四元数環">四元数環</a>の生成系に由来する。<a href="/wiki/%E7%BE%A4%E3%81%AE%E7%94%9F%E6%88%90%E5%85%83" class="mw-redirect" title="群の生成元">群の生成元</a>を </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\mapsto {\begin{bmatrix}{\sqrt {-1}}&0\\0&-{\sqrt {-1}}\end{bmatrix}},\quad j\mapsto {\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\quad k\mapsto {\begin{bmatrix}0&{\sqrt {-1}}\\{\sqrt {-1}}&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>j</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\mapsto {\begin{bmatrix}{\sqrt {-1}}&0\\0&-{\sqrt {-1}}\end{bmatrix}},\quad j\mapsto {\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\quad k\mapsto {\begin{bmatrix}0&{\sqrt {-1}}\\{\sqrt {-1}}&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2569e3ac1bd7a54a2778d5a47c2e2741e5fdf613" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:62.681ex; height:6.509ex;" alt="{\displaystyle i\mapsto {\begin{bmatrix}{\sqrt {-1}}&0\\0&-{\sqrt {-1}}\end{bmatrix}},\quad j\mapsto {\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\quad k\mapsto {\begin{bmatrix}0&{\sqrt {-1}}\\{\sqrt {-1}}&0\end{bmatrix}}}"></span></dd></dl> <p>のように対応させることで、<a href="/wiki/%E5%BF%A0%E5%AE%9F%E8%A1%A8%E7%8F%BE" title="忠実表現">忠実な行列表現</a>を得ることができる。四元数群は<a href="/w/index.php?title=%E3%83%8F%E3%83%9F%E3%83%AB%E3%83%88%E3%83%B3%E7%BE%A4&action=edit&redlink=1" class="new" title="「ハミルトン群」 (存在しないページ)">ハミルトン群</a>、つまり、すべての部分群が<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>であるような非可換群の最小位数の例である。また<a href="/wiki/%E8%87%AA%E5%B7%B1%E5%90%8C%E5%9E%8B%E7%BE%A4" class="mw-redirect" title="自己同型群">自己同型群</a> <span lang="en" class="texhtml">Aut(<i>Q</i><sub>8</sub>)</span> は4次の<a href="/wiki/%E5%AF%BE%E7%A7%B0%E7%BE%A4" title="対称群">対称群</a> <span lang="en" class="texhtml"><i>S</i><sub>4</sub></span> と同型である<sup id="cite_ref-FOOTNOTEWeinstein1977_9-0" class="reference"><a href="#cite_note-FOOTNOTEWeinstein1977-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>。 </p> <div class="mw-heading mw-heading2"><h2 id="ブラウアー・鈴木の定理"><span id=".E3.83.96.E3.83.A9.E3.82.A6.E3.82.A2.E3.83.BC.E3.83.BB.E9.88.B4.E6.9C.A8.E3.81.AE.E5.AE.9A.E7.90.86"></span>ブラウアー・鈴木の定理</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&action=edit&section=2" title="節を編集: ブラウアー・鈴木の定理"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>有限群 <span lang="en" class="texhtml mvar" style="font-style:italic;">G</span> の持つ<a href="/wiki/%E3%82%B7%E3%83%AD%E3%83%BC%E9%83%A8%E5%88%86%E7%BE%A4" class="mw-redirect" title="シロー部分群">シロー <span lang="en" class="texhtml">2</span> 部分群</a>が一般四元数群と同型ならば最大の奇数位数正規部部分群 <a href="/wiki/%E6%A0%B8_(%E7%BE%A4%E8%AB%96)" title="核 (群論)"><span lang="en" class="texhtml"><i>O</i>(<i>G</i>)</span></a> による商 <span lang="en" class="texhtml"><i>G</i>/<i>O</i>(<i>G</i>)</span> の<a href="/wiki/%E7%BE%A4%E3%81%AE%E4%B8%AD%E5%BF%83" title="群の中心">中心</a>は位数 <span lang="en" class="texhtml">2</span> である<sup id="cite_ref-FOOTNOTEMichler2006265_10-0" class="reference"><a href="#cite_note-FOOTNOTEMichler2006265-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>。特に、このような有限群 <span lang="en" class="texhtml mvar" style="font-style:italic;">G</span> は決して<a href="/wiki/%E5%8D%98%E7%B4%94%E7%BE%A4" title="単純群">単純群</a>でない。 </p> <div class="mw-heading mw-heading2"><h2 id="脚注"><span id=".E8.84.9A.E6.B3.A8"></span>脚注</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&action=edit&section=3" title="節を編集: 脚注"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="-moz-column-count:2; -webkit-column-count:2; column-count:2; -moz-column-width: 20em; -webkit-column-width: 20em; column-width: 20em; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-FOOTNOTE森197563-1">^ <a href="#cite_ref-FOOTNOTE森197563_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTE森197563_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-FOOTNOTE森197563_1-2"><sup><i><b>c</b></i></sup></a> <span class="reference-text"><a href="#CITEREF森1975">森 1975</a>, p. 63.</span> </li> <li id="cite_note-FOOTNOTE岩波数学辞典20071530-2">^ <a href="#cite_ref-FOOTNOTE岩波数学辞典20071530_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-FOOTNOTE岩波数学辞典20071530_2-1"><sup><i><b>b</b></i></sup></a> <span class="reference-text"><a href="#CITEREF岩波数学辞典2007">岩波数学辞典 2007</a>, p. 1530.</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"><a href="#CITEREF近藤1991">近藤 (1991</a>, p. 31)は「4元数型の群」、<a href="#CITEREF鈴木1977">鈴木 (1977</a>, p. 255)は「4元数形の2群」という言い方をしている。 </span> </li> <li id="cite_note-FOOTNOTEATLAS1985xx-4"><b><a href="#cite_ref-FOOTNOTEATLAS1985xx_4-0">^</a></b> <span class="reference-text"><a href="#CITEREFATLAS1985">ATLAS 1985</a>, p. xx.</span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">「2重巡回群」（<a href="/wiki/%E8%8B%B1%E8%AA%9E" title="英語">英</a>: <span lang="en">dicyclic group</span>）と呼ばれることもある (<a href="#CITEREFアームストロング2007">アームストロング 2007</a>, p. 195)。</span> </li> <li id="cite_note-FOOTNOTE近藤199131,_382-6"><b><a href="#cite_ref-FOOTNOTE近藤199131,_382_6-0">^</a></b> <span class="reference-text"><a href="#CITEREF近藤1991">近藤 1991</a>, pp. 31, 382.</span> </li> <li id="cite_note-FOOTNOTE鈴木1977255-7"><b><a href="#cite_ref-FOOTNOTE鈴木1977255_7-0">^</a></b> <span class="reference-text"><a href="#CITEREF鈴木1977">鈴木 1977</a>, p. 255.</span> </li> <li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text">一般四元数群の対応する表示は <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 Q_{4m}=\langle \,i,j,k\mid i^{m}=j^{2}=k^{2}=ijk\,\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> <mo>∣</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>i</mi> <mi>j</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{4m}=\langle \,i,j,k\mid i^{m}=j^{2}=k^{2}=ijk\,\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e88e0eb25d8a5ec99c2b9513372baac65bfa521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.017ex; height:3.176ex;" alt="{\displaystyle Q_{4m}=\langle \,i,j,k\mid i^{m}=j^{2}=k^{2}=ijk\,\rangle }"></span> である (<a href="#Groupprops">Groupprops</a>)。</span> </li> <li id="cite_note-FOOTNOTEWeinstein1977-9"><b><a href="#cite_ref-FOOTNOTEWeinstein1977_9-0">^</a></b> <span class="reference-text"><a href="#CITEREFWeinstein1977">Weinstein 1977</a>.</span> </li> <li id="cite_note-FOOTNOTEMichler2006265-10"><b><a href="#cite_ref-FOOTNOTEMichler2006265_10-0">^</a></b> <span class="reference-text"><a href="#CITEREFMichler2006">Michler 2006</a>, p. <span class="plainlinks"><a rel="nofollow" class="external text" href="https://books.google.co.jp/books?id=lWfF-NHeq6cC&pg=PA265">265</a></span>.</span> </li> </ol></div> <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=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&action=edit&section=4" title="節を編集: 参考文献"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREFアームストロング2007">M. A. アームストロング『対称性からの群論入門』<a href="/wiki/%E3%82%B7%E3%83%A5%E3%83%97%E3%83%AA%E3%83%B3%E3%82%AC%E3%83%BC%E3%83%BB%E3%82%B8%E3%83%A3%E3%83%91%E3%83%B3" title="シュプリンガー・ジャパン">シュプリンガー・ジャパン</a>、2007年。<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><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%99/978-4-431-10007-2" title="特別:文献資料/978-4-431-10007-2">978-4-431-10007-2</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=%E5%AF%BE%E7%A7%B0%E6%80%A7%E3%81%8B%E3%82%89%E3%81%AE%E7%BE%A4%E8%AB%96%E5%85%A5%E9%96%80&rft.aulast=M.+A.+%E3%82%A2%E3%83%BC%E3%83%A0%E3%82%B9%E3%83%88%E3%83%AD%E3%83%B3%E3%82%B0&rft.au=M.+A.+%E3%82%A2%E3%83%BC%E3%83%A0%E3%82%B9%E3%83%88%E3%83%AD%E3%83%B3%E3%82%B0&rft.date=2007&rft.pub=%5B%5B%E3%82%B7%E3%83%A5%E3%83%97%E3%83%AA%E3%83%B3%E3%82%AC%E3%83%BC%E3%83%BB%E3%82%B8%E3%83%A3%E3%83%91%E3%83%B3%5D%5D&rft.isbn=978-4-431-10007-2&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREF近藤1991">近藤武『群論』<a href="/wiki/%E5%B2%A9%E6%B3%A2%E6%9B%B8%E5%BA%97" title="岩波書店">岩波書店</a>〈<a href="/wiki/%E5%B2%A9%E6%B3%A2%E5%9F%BA%E7%A4%8E%E6%95%B0%E5%AD%A6%E9%81%B8%E6%9B%B8" class="mw-redirect" title="岩波基礎数学選書">岩波基礎数学選書</a>〉、1991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%99/4-00-007807-0" title="特別:文献資料/4-00-007807-0">4-00-007807-0</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=%E7%BE%A4%E8%AB%96&rft.aulast=%E8%BF%91%E8%97%A4%E6%AD%A6&rft.au=%E8%BF%91%E8%97%A4%E6%AD%A6&rft.date=1991&rft.series=%5B%5B%E5%B2%A9%E6%B3%A2%E5%9F%BA%E7%A4%8E%E6%95%B0%E5%AD%A6%E9%81%B8%E6%9B%B8%5D%5D&rft.pub=%5B%5B%E5%B2%A9%E6%B3%A2%E6%9B%B8%E5%BA%97%5D%5D&rft.isbn=4-00-007807-0&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREF鈴木1977"><a href="/wiki/%E9%88%B4%E6%9C%A8%E9%80%9A%E5%A4%AB" title="鈴木通夫">鈴木通夫</a>『群論　上』 18巻、<a href="/wiki/%E5%B2%A9%E6%B3%A2%E6%9B%B8%E5%BA%97" title="岩波書店">岩波書店</a>〈現代数学〉、1977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%99/978-4-00-730271-8" title="特別:文献資料/978-4-00-730271-8">978-4-00-730271-8</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=%E7%BE%A4%E8%AB%96%E3%80%80%E4%B8%8A&rft.aulast=%E9%88%B4%E6%9C%A8%E9%80%9A%E5%A4%AB&rft.au=%E9%88%B4%E6%9C%A8%E9%80%9A%E5%A4%AB&rft.date=1977&rft.series=%E7%8F%BE%E4%BB%A3%E6%95%B0%E5%AD%A6&rft.pub=%5B%5B%E5%B2%A9%E6%B3%A2%E6%9B%B8%E5%BA%97%5D%5D&rft.isbn=978-4-00-730271-8&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span> 数学 <a rel="nofollow" class="external text" href="https://doi.org/10.11429/sugaku1947.37.180">sugaku1947.37.180</a></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREF森1975">森光弥『有限群論入門』<a href="/wiki/%E5%AE%9F%E6%95%99%E5%87%BA%E7%89%88" title="実教出版">実教出版</a>、1975年。<a href="/wiki/%E5%9B%BD%E7%AB%8B%E5%9B%BD%E4%BC%9A%E5%9B%B3%E6%9B%B8%E9%A4%A8#書誌データの提供" title="国立国会図書館">全国書誌番号</a>:<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a rel="nofollow" class="external text" href="//id.ndl.go.jp/jpno/69006293">69006293</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=%E6%9C%89%E9%99%90%E7%BE%A4%E8%AB%96%E5%85%A5%E9%96%80&rft.aulast=%E6%A3%AE%E5%85%89%E5%BC%A5&rft.au=%E6%A3%AE%E5%85%89%E5%BC%A5&rft.date=1975&rft.pub=%5B%5B%E5%AE%9F%E6%95%99%E5%87%BA%E7%89%88%5D%5D&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREF岩波数学辞典2007"><a href="/wiki/%E6%97%A5%E6%9C%AC%E6%95%B0%E5%AD%A6%E4%BC%9A" title="日本数学会">日本数学会</a> 編『岩波数学辞典』（第4版）<a href="/wiki/%E5%B2%A9%E6%B3%A2%E6%9B%B8%E5%BA%97" title="岩波書店">岩波書店</a>、2007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%99/978-4-00-080309-0" title="特別:文献資料/978-4-00-080309-0">978-4-00-080309-0</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=%E5%B2%A9%E6%B3%A2+%E6%95%B0%E5%AD%A6%E8%BE%9E%E5%85%B8&rft.date=2007&rft.edition=%E7%AC%AC4%E7%89%88&rft.pub=%5B%5B%E5%B2%A9%E6%B3%A2%E6%9B%B8%E5%BA%97%5D%5D&rft.isbn=978-4-00-080309-0&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREFMichler2006">Michler, Gerhard O. (2006). <i>Theory of Finite Simple Groups</i>. New Mathematical Monographs. <b>8</b>. <a href="/wiki/Cambridge_University_Press" class="mw-redirect" title="Cambridge University Press">Cambridge University Press</a>. <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%99/0-521-86625-1" title="特別:文献資料/0-521-86625-1">0-521-86625-1</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="/wiki/Zentralblatt_MATH" class="mw-redirect" title="Zentralblatt MATH">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1146.20011">1146.20011</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=Theory+of+Finite+Simple+Groups&rft.aulast=Michler&rft.aufirst=Gerhard+O.&rft.au=Michler%2C%26%2332%3BGerhard+O.&rft.date=2006&rft.series=New+Mathematical+Monographs&rft.volume=8&rft.pub=%5B%5BCambridge+University+Press%5D%5D&rft.isbn=0-521-86625-1&rft_id=info:zbl/1146.20011&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREFATLAS1985">Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A. (1985). <i>Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups</i>. Oxford. <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%99/0-19-853199-0" title="特別:文献資料/0-19-853199-0">0-19-853199-0</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="/wiki/Zentralblatt_MATH" class="mw-redirect" title="Zentralblatt MATH">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0568.20001">0568.20001</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=Atlas+of+Finite+Groups%3A+Maximal+Subgroups+and+Ordinary+Characters+for+Simple+Groups&rft.aulast=Conway&rft.aufirst=J.+H.&rft.au=Conway%2C%26%2332%3BJ.+H.&rft.au=Curtis%2C%26%2332%3BR.+T.&rft.au=Norton%2C%26%2332%3BS.+P.&rft.au=Parker%2C%26%2332%3BR.+A.&rft.au=Wilson%2C%26%2332%3BR.+A.&rft.date=1985&rft.pub=Oxford&rft.isbn=0-19-853199-0&rft_id=info:zbl/0568.20001&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <ul><li><cite style="font-style:normal" class="citation book" id="CITEREFWeinstein1977">Weinstein, M. (1977). <i>Example of Groups</i>. Polygonal Publishing House. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r101121245"><a href="/wiki/Zentralblatt_MATH" class="mw-redirect" title="Zentralblatt MATH">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0359.20001">0359.20001</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=Example+of+Groups&rft.aulast=Weinstein&rft.aufirst=M.&rft.au=Weinstein%2C%26%2332%3BM.&rft.date=1977&rft.pub=Polygonal+Publishing+House&rft_id=info:zbl/0359.20001&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部リンク"><span id=".E5.A4.96.E9.83.A8.E3.83.AA.E3.83.B3.E3.82.AF"></span>外部リンク</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4&action=edit&section=5" title="節を編集: 外部リンク"><span>編集</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><cite style="font-style:normal" class="citation" id="CITEREFHazewinkel2001">Hazewinkel, Michiel, ed. (2001), <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Quaternion_group">“Quaternion group”</a>, <i><a href="https://en.wikipedia.org/wiki/Encyclopedia_of_Mathematics" class="extiw" title="en:Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer, <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%99/978-1-55608-010-4" title="特別:文献資料/978-1-55608-010-4">978-1-55608-010-4</a><span style="display:none;">, <a rel="nofollow" class="external free" href="https://www.encyclopediaofmath.org/index.php?title=Quaternion_group">https://www.encyclopediaofmath.org/index.php?title=Quaternion_group</a></span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.btitle=Quaternion+group&rft.atitle=%5B%5B%3Aen%3AEncyclopedia+of+Mathematics%7CEncyclopedia+of+Mathematics%5D%5D&rft.date=2001&rft.pub=Springer&rft.isbn=978-1-55608-010-4&rft_id=&rfr_id=info:sid/ja.wikipedia.org:%E4%B8%80%E8%88%AC%E5%9B%9B%E5%85%83%E6%95%B0%E7%BE%A4"><span style="display: none;"> </span></span></li> <li><cite class="citation web" style="font-style:normal" id="Groupprops">“<a rel="nofollow" class="external text" href="https://groupprops.subwiki.org/wiki/Dicyclic_group">Dicyclic group</a>”. <i>Groupprops</i>. <span title="">2019年5月26日</span>閲覧。</cite></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" 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