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[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Đăng nhập</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"> Trang dành cho người dùng chưa đăng nhập <a href="/wiki/Tr%E1%BB%A3_gi%C3%BAp:Gi%E1%BB%9Bi_thi%E1%BB%87u" aria-label="Tìm hiểu thêm về sửa đổi"><span>tìm hiểu thêm</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/%C4%90%E1%BA%B7c_bi%E1%BB%87t:%C4%90%C3%B3ng_g%C3%B3p_c%E1%BB%A7a_t%C3%B4i" title="Danh sách các sửa đổi được thực hiện qua địa chỉ IP này [y]" accesskey="y"><span>Đóng góp</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%C4%90%E1%BA%B7c_bi%E1%BB%87t:Th%E1%BA%A3o_lu%E1%BA%ADn_t%C3%B4i" title="Thảo luận với địa chỉ IP này [n]" accesskey="n"><span>Thảo luận cho địa chỉ IP 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class="vector-pinnable-header-label">Nội dung</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ẩn</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">Đầu</div> </a> </li> <li id="toc-Ví_dụ" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ví_dụ"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Ví dụ</span> </div> </a> <ul id="toc-Ví_dụ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cấp_và_cấu_trúc_của_nhóm" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Cấp_và_cấu_trúc_của_nhóm"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Cấp và cấu trúc của nhóm</span> </div> </a> <button aria-controls="toc-Cấp_và_cấu_trúc_của_nhóm-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Hiện/ẩn mục Cấp và cấu trúc của nhóm</span> </button> <ul id="toc-Cấp_và_cấu_trúc_của_nhóm-sublist" class="vector-toc-list"> <li id="toc-Mối_liên_hệ_giữa_hai_khái_niệm_của_cấp" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mối_liên_hệ_giữa_hai_khái_niệm_của_cấp"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Mối liên hệ giữa hai khái niệm của cấp</span> </div> </a> <ul id="toc-Mối_liên_hệ_giữa_hai_khái_niệm_của_cấp-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Định_lý_Lagrange" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Định_lý_Lagrange"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Định lý Lagrange</span> </div> </a> <ul id="toc-Định_lý_Lagrange-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Các_tính_chất_của_cấp_của_phần_tử" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Các_tính_chất_của_cấp_của_phần_tử"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Các tính chất của cấp của phần tử</span> </div> </a> <ul id="toc-Các_tính_chất_của_cấp_của_phần_tử-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Định_lý_Cauchy_(định_lý_Côsi)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Định_lý_Cauchy_(định_lý_Côsi)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Định lý Cauchy (định lý Côsi)</span> </div> </a> <ul id="toc-Định_lý_Cauchy_(định_lý_Côsi)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Đồng_cấu_nhóm_và_cấp" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Đồng_cấu_nhóm_và_cấp"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Đồng cấu nhóm và cấp</span> </div> </a> <ul id="toc-Đồng_cấu_nhóm_và_cấp-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bản_số_của_nhóm_con_chuẩn_tắc" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bản_số_của_nhóm_con_chuẩn_tắc"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Bản số của nhóm con chuẩn tắc</span> </div> </a> <ul id="toc-Bản_số_của_nhóm_con_chuẩn_tắc-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Chú_thích" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Chú_thích"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Chú thích</span> </div> </a> <ul id="toc-Chú_thích-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tham_khảo" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tham_khảo"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Tham khảo</span> </div> </a> <ul id="toc-Tham_khảo-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Liên_kết_ngoài" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Liên_kết_ngoài"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Liên kết ngoài</span> </div> </a> <ul id="toc-Liên_kết_ngoài-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="Nội dung" 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="Đóng mở mục lục" > <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">Đóng mở mục lục</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">Cấp (lý thuyết nhóm)</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="Xem bài viết trong ngôn ngữ khác. Bài có sẵn trong 19 ngôn ngữ" > <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-19" 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">19 ngôn ngữ</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/%D9%85%D8%B1%D8%AA%D8%A8%D8%A9_(%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%B2%D9%85%D8%B1)" title="مرتبة (نظرية الزمر) – Tiếng Ả Rập" lang="ar" hreflang="ar" data-title="مرتبة (نظرية الزمر)" data-language-autonym="العربية" data-language-local-name="Tiếng Ả Rập" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/%C5%98%C3%A1d_prvku" title="Řád prvku – Tiếng Séc" lang="cs" hreflang="cs" data-title="Řád prvku" data-language-autonym="Čeština" data-language-local-name="Tiếng Séc" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Order_(group_theory)" title="Order (group theory) – Tiếng Anh" lang="en" hreflang="en" data-title="Order (group theory)" data-language-autonym="English" data-language-local-name="Tiếng Anh" 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/Orden_(teor%C3%ADa_de_grupos)" title="Orden (teoría de grupos) – Tiếng Tây Ban Nha" lang="es" hreflang="es" data-title="Orden (teoría de grupos)" data-language-autonym="Español" data-language-local-name="Tiếng Tây Ban Nha" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Ordre_(th%C3%A9orie_des_groupes)" title="Ordre (théorie des groupes) – Tiếng Pháp" lang="fr" hreflang="fr" data-title="Ordre (théorie des groupes)" data-language-autonym="Français" data-language-local-name="Tiếng Pháp" 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/Orde_(teor%C3%ADa_de_grupos)" title="Orde (teoría de grupos) – Tiếng Galician" lang="gl" hreflang="gl" data-title="Orde (teoría de grupos)" data-language-autonym="Galego" data-language-local-name="Tiếng Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko badge-Q70893996 mw-list-item" title=""><a href="https://ko.wikipedia.org/wiki/%EC%9C%84%EC%88%98_(%EC%88%98%ED%95%99)" title="위수 (수학) – Tiếng Hàn" lang="ko" hreflang="ko" data-title="위수 (수학)" data-language-autonym="한국어" data-language-local-name="Tiếng Hàn" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A1%D7%93%D7%A8_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="סדר (תורת החבורות) – Tiếng Do Thái" lang="he" hreflang="he" data-title="סדר (תורת החבורות)" data-language-autonym="עברית" data-language-local-name="Tiếng Do Thái" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B5%8B%E0%B4%9F%E0%B4%BF_(%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D_%E0%B4%B8%E0%B4%BF%E0%B4%A6%E0%B5%8D%E0%B4%A7%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%82)" title="കോടി (ഗ്രൂപ്പ് സിദ്ധാന്തം) – Tiếng Malayalam" lang="ml" hreflang="ml" data-title="കോടി (ഗ്രൂപ്പ് സിദ്ധാന്തം)" data-language-autonym="മലയാളം" data-language-local-name="Tiếng Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Orde_(groepentheorie)" title="Orde (groepentheorie) – Tiếng Hà Lan" lang="nl" hreflang="nl" data-title="Orde (groepentheorie)" data-language-autonym="Nederlands" data-language-local-name="Tiếng Hà Lan" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BD%8D%E6%95%B0_(%E7%BE%A4%E8%AB%96)" title="位数 (群論) – Tiếng Nhật" lang="ja" hreflang="ja" data-title="位数 (群論)" data-language-autonym="日本語" data-language-local-name="Tiếng Nhật" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rz%C4%85d_(teoria_grup)" title="Rząd (teoria grup) – Tiếng Ba Lan" lang="pl" hreflang="pl" data-title="Rząd (teoria grup)" data-language-autonym="Polski" data-language-local-name="Tiếng Ba Lan" 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/Ordem_(teoria_dos_grupos)" title="Ordem (teoria dos grupos) – Tiếng Bồ Đào Nha" lang="pt" hreflang="pt" data-title="Ordem (teoria dos grupos)" data-language-autonym="Português" data-language-local-name="Tiếng Bồ Đào Nha" 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/Ordin_(teoria_grupurilor)" title="Ordin (teoria grupurilor) – Tiếng Romania" lang="ro" hreflang="ro" data-title="Ordin (teoria grupurilor)" data-language-autonym="Română" data-language-local-name="Tiếng Romania" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Order_(group_theory)" title="Order (group theory) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Order (group theory)" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ordning_(gruppteori)" title="Ordning (gruppteori) – Tiếng Thụy Điển" lang="sv" hreflang="sv" data-title="Ordning (gruppteori)" data-language-autonym="Svenska" data-language-local-name="Tiếng Thụy Điển" 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%95%E0%AF%81%E0%AE%B2%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%95%E0%AE%BF%E0%AE%B0%E0%AE%AE%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="குலத்தின் கிரமம் (கணிதம்) – Tiếng Tamil" lang="ta" hreflang="ta" data-title="குலத்தின் கிரமம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tiếng Tamil" 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%9F%D0%BE%D1%80%D1%8F%D0%B4%D0%BE%D0%BA_(%D1%82%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B3%D1%80%D1%83%D0%BF)" title="Порядок (теорія груп) – Tiếng Ukraina" lang="uk" hreflang="uk" data-title="Порядок (теорія груп)" data-language-autonym="Українська" data-language-local-name="Tiếng Ukraina" 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/%E9%9A%8E_(%E7%BE%A4%E8%AB%96)" title="階 (群論) – Tiếng Trung" lang="zh" hreflang="zh" data-title="階 (群論)" data-language-autonym="中文" data-language-local-name="Tiếng Trung" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q589491#sitelinks-wikipedia" title="Sửa liên kết giữa ngôn ngữ" class="wbc-editpage">Sửa liên kết</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="Không gian tên"> <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 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</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="Công cụ trang"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Giao diện"> <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">Giao diện</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">chuyển sang thanh bên</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">ẩn</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">Bách khoa toàn thư mở 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="vi" dir="ltr"><p>Trong <a href="/wiki/L%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m" title="Lý thuyết nhóm">lý thuyết nhóm</a>, thuật ngữ <b>cấp</b> (tiếng Anh: <i>order</i>) có hai ý nghĩa, cả hai ý nghĩa này đều liên hệ mật thiết với nhau: </p> <ul><li>cấp của một <a href="/wiki/Nh%C3%B3m_(%C4%91%E1%BA%A1i_s%E1%BB%91)" class="mw-redirect" title="Nhóm (đại số)">nhóm</a> <i>G</i> chính là số phần tử của <i>G</i>;<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></li> <li>cấp của phần tử <i>a</i> trong nhóm <i>G</i> là số nguyên dương <i>m</i> nhỏ nhất thỏa mãn <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^{m}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{m}=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620a4abed0abe81d092c5fb9fba79403a38d7e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.087ex; height:2.343ex;" alt="{\displaystyle a^{m}=e}"></span>, trong đó <i>e</i> là <a href="/wiki/Ph%E1%BA%A7n_t%E1%BB%AD_%C4%91%C6%A1n_v%E1%BB%8B" title="Phần tử đơn vị">phần tử đơn vị</a> của nhóm <i>G</i>, <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^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba5a8f0d974fc35561ad3590523c142123cd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.905ex; height:2.343ex;" alt="{\displaystyle a^{m}}"></span> là tích (với phép toán trang bị cho nhóm <i>G</i>) của <i>m</i> phần tử <i>a</i>.</li></ul> <p>Ký hiệu cấp của nhóm <i>G</i> là <i>ord(G)</i> hoặc <i>|G|</i>; cấp của phần tử <i>a</i> được ký hiệu là <i>ord(a)</i> hoặc <i>|a|</i>. </p><p>Cấp của nhóm và của phần tử có thể hữu hạn hoặc vô hạn ∞. Ví dụ tập hợp các số nguyên <i>Z</i> cùng với phép toán cộng + lập thành một nhóm có cấp bằng ∞ (vì <i>Z</i> có vô số phần tử). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Ví_dụ"><span id="V.C3.AD_d.E1.BB.A5"></span>Ví dụ</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=1" title="Sửa đổi phần “Ví dụ”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=1" title="Sửa mã nguồn tại đề mục: Ví dụ"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bảng sau là bảng nhân cho các phần tử của <a href="/wiki/Nh%C3%B3m_%C4%91%E1%BB%91i_x%E1%BB%A9ng" class="mw-redirect" title="Nhóm đối xứng">nhóm đối xứng</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e15f3e200aaa247f69c43110cc5a09ecc91b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}"></span>: </p> <dl><dd><table cellspacing="0" cellpadding="8" border="1"> <tbody><tr> <th>• </th> <th><i>e</i></th> <th><i>s</i></th> <th><i>t</i></th> <th><i>u</i></th> <th><i>v</i></th> <th><i>w</i> </th></tr> <tr> <th><i>e</i> </th> <td><span style="color:#009246"><i>e</i></span></td> <td><i>s</i></td> <td><i>t</i></td> <td><i>u</i></td> <td><i>v</i></td> <td><i>w</i> </td></tr> <tr> <th><i>s</i> </th> <td><i>s</i></td> <td><span style="color:#009246"><i>e</i></span></td> <td><i>v</i></td> <td><i>w</i></td> <td><i>t</i></td> <td><i>u</i> </td></tr> <tr> <th><i>t</i> </th> <td><i>t</i></td> <td><i>u</i></td> <td><span style="color:#009246"><i>e</i></span></td> <td><i>s</i></td> <td><i>w</i></td> <td><i>v</i> </td></tr> <tr> <th><i>u</i> </th> <td><i>u</i></td> <td><i>t</i></td> <td><i>w</i></td> <td><span style="color:#009246"><i>v</i></span></td> <td><i>e</i></td> <td><i>s</i> </td></tr> <tr> <th><i>v</i> </th> <td><i>v</i></td> <td><i>w</i></td> <td><i>s</i></td> <td><i>e</i></td> <td><span style="color:#009246"><i>u</i></span></td> <td><i>t</i> </td></tr> <tr> <th><i>w</i> </th> <td><i>w</i></td> <td><i>v</i></td> <td><i>u</i></td> <td><i>t</i></td> <td><i>s</i></td> <td><span style="color:#009246"><i>e</i></span> </td></tr></tbody></table></dd></dl> <p>Nhóm <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 S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e15f3e200aaa247f69c43110cc5a09ecc91b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}"></span> có 6 phần tử, nên cấp của nó bằng 6: </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 ord(S_{3})=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mi>r</mi> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ord(S_{3})=6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ff100cb279123ff3f0298825025b4ced5f9a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.942ex; height:2.843ex;" alt="{\displaystyle ord(S_{3})=6}"></span>;</dd></dl> <p>Cấp của các phần tử trong nhóm <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 S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e15f3e200aaa247f69c43110cc5a09ecc91b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}"></span>: </p> <ul><li>phần tử đơn vị <i>e</i> có cấp bằng 1;</li> <li>các phần tử <i>s</i>, <i>t</i>, <i>w</i> <a href="/wiki/B%C3%ACnh_ph%C6%B0%C6%A1ng" title="Bình phương">bình phương</a> lên bằng <i>e</i>: <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 s^{2}=t^{2}=w^{2}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}=t^{2}=w^{2}=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e3d0d0df5dfd7e6df069617e51cef5868792183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.136ex; height:2.676ex;" alt="{\displaystyle s^{2}=t^{2}=w^{2}=e}"></span>, nên chúng có cấp bằng 2;</li> <li>các phần tử <i>u</i> và <i>v</i> có cấp bằng 3; điều này có thể giải thích như sau: <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 u^{2}=v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{2}=v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9c2a3f2edbcc94a1954f847c3eb29bbf742df0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.61ex; height:2.676ex;" alt="{\displaystyle u^{2}=v}"></span> nên <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 u^{3}=uv=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mi>u</mi> <mi>v</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u^{3}=uv=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d57c4353ef8a798d4e76cf66a3698d80b5f578b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.122ex; height:2.676ex;" alt="{\displaystyle u^{3}=uv=e}"></span>, tương tự cho v..</li></ul> <div class="mw-heading mw-heading2"><h2 id="Cấp_và_cấu_trúc_của_nhóm"><span id="C.E1.BA.A5p_v.C3.A0_c.E1.BA.A5u_tr.C3.BAc_c.E1.BB.A7a_nh.C3.B3m"></span>Cấp và cấu trúc của nhóm</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=2" title="Sửa đổi phần “Cấp và cấu trúc của nhóm”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=2" title="Sửa mã nguồn tại đề mục: Cấp và cấu trúc của nhóm"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cấp của nhóm và cấp của phần tử trong nhóm nói lên rất nhiều điều về cấu trúc của chính nhóm đó. </p><p>Nếu cấp của nhóm <i>G</i> bằng 1 thì nó là <a href="/w/index.php?title=Nh%C3%B3m_t%E1%BA%A7m_th%C6%B0%E1%BB%9Dng&action=edit&redlink=1" class="new" title="Nhóm tầm thường (trang không tồn tại)">nhóm tầm thường</a>. </p><p>Nếu cấp của phần tử <i>a</i> bằng 1: <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^{1}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{1}=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4ce964ff1032dc0d2c77ac712151c526444c15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.466ex; height:2.676ex;" alt="{\displaystyle a^{1}=e}"></span> thì <i>a</i> chính là phần tử đơn vị của nhóm. </p><p>Nếu mọi phần tử <i>a</i> (khác phần tử đơn vị) của nhóm <i>G</i> đều bằng <a href="/w/index.php?title=Ngh%E1%BB%8Bch_%C4%91%E1%BA%A3o&action=edit&redlink=1" class="new" title="Nghịch đảo (trang không tồn tại)">nghịch đảo</a> của chính các phần tử đó (<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^{2}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8538895139f85f93d237b712874ab22bb5e7d725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.466ex; height:2.676ex;" alt="{\displaystyle a^{2}=e}"></span>) thì chúng đều có cấp bằng 2: <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 ord(a)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mi>r</mi> <mi>d</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ord(a)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aedd6487a074ad56326d367b381b9f05877740a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.692ex; height:2.843ex;" alt="{\displaystyle ord(a)=2}"></span> và nhóm <i>G</i> là nhóm Abel, vì: </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 ab=(bb)ab(aa)=b(ba)(ba)a=b(ba)^{2}a=bea=ba}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>e</mi> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=(bb)ab(aa)=b(ba)(ba)a=b(ba)^{2}a=bea=ba}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a23bca126afec0bc59f093ef50699162935007d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.178ex; height:3.176ex;" alt="{\displaystyle ab=(bb)ab(aa)=b(ba)(ba)a=b(ba)^{2}a=bea=ba}"></span>.</dd></dl> <p>Điều ngược lại chưa chắc đúng. Ví dụ <a href="/w/index.php?title=Nh%C3%B3m_c%E1%BB%99ng_c%C3%A1c_s%E1%BB%91_nguy%C3%AAn_modulo_6&action=edit&redlink=1" class="new" title="Nhóm cộng các số nguyên modulo 6 (trang không tồn tại)">nhóm cộng các số nguyên modulo</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f10f6f62aa32dbac6ff8f700668764d87fb840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{6}}"></span> là nhóm Abel, nhưng không phải mọi phần tử của nó đều có cấp bằng 2, ví dụ phần tử 2 có cấp bằng 3: <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 2+2+2=0{\pmod {6}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>6</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+2+2=0{\pmod {6}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d6c5c9bc13cd173f22ea891a22c59dfbe9603a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.276ex; height:2.843ex;" alt="{\displaystyle 2+2+2=0{\pmod {6}}}"></span>. </p><p>Một phần tử <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> có cấp bằng <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 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"></span> cũng được gọi là một phần tử <a href="/wiki/L%C5%A9y_%C4%91%E1%BA%B3ng" class="mw-redirect" title="Lũy đẳng">lũy đẳng</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Mối_liên_hệ_giữa_hai_khái_niệm_của_cấp"><span id="M.E1.BB.91i_li.C3.AAn_h.E1.BB.87_gi.E1.BB.AFa_hai_kh.C3.A1i_ni.E1.BB.87m_c.E1.BB.A7a_c.E1.BA.A5p"></span>Mối liên hệ giữa hai khái niệm của cấp</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=3" title="Sửa đổi phần “Mối liên hệ giữa hai khái niệm của cấp”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=3" title="Sửa mã nguồn tại đề mục: Mối liên hệ giữa hai khái niệm của cấp"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mối liên hệ giữa hai khái niệm của cấp được giải thích như sau: </p> <dl><dd>nếu ta lấy <a href="/wiki/Nh%C3%B3m_cyclic" title="Nhóm cyclic">nhóm xyclic</a> sinh bởi phần tử <i>g</i>, ký hiệu là <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 \langle g\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle g\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0617219bf579b828e25823da330be038f4f5fe37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.925ex; height:2.843ex;" alt="{\displaystyle \langle g\rangle }"></span>: <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 \langle g\rangle =\{g^{m}|m\in \mathbb {Z} \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle g\rangle =\{g^{m}|m\in \mathbb {Z} \}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69b6967273f2753e5781e673ebde6530bdd61479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.22ex; height:2.843ex;" alt="{\displaystyle \langle g\rangle =\{g^{m}|m\in \mathbb {Z} \}}"></span>,</dd></dl></dd> <dd>thì cấp của nhóm <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 \langle g\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle g\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0617219bf579b828e25823da330be038f4f5fe37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.925ex; height:2.843ex;" alt="{\displaystyle \langle g\rangle }"></span> chính bằng cấp của phần tử <i>g</i>: <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 \operatorname {ord} (\langle g\rangle )=\operatorname {ord} (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>g</mi> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} (\langle g\rangle )=\operatorname {ord} (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835cb637186095aa4fc0531944c9f4fc01796d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.492ex; height:2.843ex;" alt="{\displaystyle \operatorname {ord} (\langle g\rangle )=\operatorname {ord} (g)}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Định_lý_Lagrange"><span id=".C4.90.E1.BB.8Bnh_l.C3.BD_Lagrange"></span>Định lý Lagrange</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=4" title="Sửa đổi phần “Định lý Lagrange”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=4" title="Sửa mã nguồn tại đề mục: Định lý Lagrange"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="hatnote navigation-not-searchable">Bài chi tiết: <a href="/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Lagrange_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)" title="Định lý Lagrange (lý thuyết nhóm)">Định lý Lagrange (lý thuyết nhóm)</a></div> <p><b>Định lý Lagrange</b>: Nếu <i>H</i> là <a href="/wiki/Nh%C3%B3m_con" title="Nhóm con">nhóm con</a> của <i>G</i>, và <i>G</i> có hữu hạn phần tử, thì <i>H</i> cũng hữu hạn và có cấp là ước số của cấp của <i>G</i>: </p> <dl><dd><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 {\frac {\operatorname {ord} (G)}{\operatorname {ord} (H)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\operatorname {ord} (G)}{\operatorname {ord} (H)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce64e601f5840ee1de832c0d0b1da5706a47459d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:8.076ex; height:6.509ex;" alt="{\displaystyle {\frac {\operatorname {ord} (G)}{\operatorname {ord} (H)}}}"></span> là số tự nhiên và bằng bản số của nhóm thương (<i>G</i>:<i>H</i>)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>.</dd></dl></dd></dl> <p>Từ định lý trên có thể suy ra, cấp của <i>G</i> chia hết cho cấp của mọi phần từ <i>a</i> thuộc <i>G</i>. Như đã xét trong ví dụ về nhóm <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 S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e15f3e200aaa247f69c43110cc5a09ecc91b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}"></span>, ord(<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 S_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e15f3e200aaa247f69c43110cc5a09ecc91b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{3}}"></span>)=6 chia hết cho 2 là cấp của <i>s</i>,<i>t</i>,<i>w</i> và 3 là cấp của <i>u</i>,<i>v</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Các_tính_chất_của_cấp_của_phần_tử"><span id="C.C3.A1c_t.C3.ADnh_ch.E1.BA.A5t_c.E1.BB.A7a_c.E1.BA.A5p_c.E1.BB.A7a_ph.E1.BA.A7n_t.E1.BB.AD"></span>Các tính chất của cấp của phần tử</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=5" title="Sửa đổi phần “Các tính chất của cấp của phần tử”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=5" title="Sửa mã nguồn tại đề mục: Các tính chất của cấp của phần tử"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Phần tử <i>a</i> và nghịch đảo của nó <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^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5709c8d86f7fec8fb86069bf5d15a9eabe564e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.563ex; height:2.676ex;" alt="{\displaystyle a^{-1}}"></span> có cùng cấp: </p> <dl><dd><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 \operatorname {ord} (a)=\operatorname {ord} (a^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} (a)=\operatorname {ord} (a^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d92a83a68367777e0f4c22e6a94cdf6f21445c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.243ex; height:3.176ex;" alt="{\displaystyle \operatorname {ord} (a)=\operatorname {ord} (a^{-1})}"></span>.</dd></dl></dd></dl> <p>Nếu số nguyên <i>k</i> thỏa mãn: <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^{k}=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{k}=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c3f47cfcbe91f000ee33947c3dcff930aa64a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.5ex; height:2.676ex;" alt="{\displaystyle a^{k}=e}"></span> thì cấp của <i>a</i> là ước của <i>k</i>. Nhận xét này được áp dụng rất nhiều trong số học sơ cấp. </p><p>Nếu <i>a</i> có cấp hữu hạn thì mọi lũy thừa nguyên của <i>a</i> cũng có cấp hữu hạn. Cấp của phần tử <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^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e30fb4fe7c63445f6f0f433e87ae65d87ff55f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.319ex; height:2.676ex;" alt="{\displaystyle a^{k}}"></span> được tính như sau: </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 \operatorname {ord} (a^{k})={\frac {\operatorname {ord} (a)}{UCLN(\operatorname {ord} (a),k)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>C</mi> <mi>L</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} (a^{k})={\frac {\operatorname {ord} (a)}{UCLN(\operatorname {ord} (a),k)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea9c03f54cfd67953272ff30d2045da3b7b6bdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.085ex; height:6.509ex;" alt="{\displaystyle \operatorname {ord} (a^{k})={\frac {\operatorname {ord} (a)}{UCLN(\operatorname {ord} (a),k)}}}"></span> (ký hiệu UCLN(<i>a</i>,<i>b</i>) là <a href="/wiki/%C6%AF%E1%BB%9Bc_s%E1%BB%91_chung_l%E1%BB%9Bn_nh%E1%BA%A5t" title="Ước số chung lớn nhất">ước số chung lớn nhất</a> của <i>a</i> và <i>b</i>).</dd></dl> <p>Ví dụ: </p> <ul><li>Trong <a href="/w/index.php?title=Nh%C3%B3m_c%E1%BB%99ng_c%C3%A1c_s%E1%BB%91_nguy%C3%AAn_modulo&action=edit&redlink=1" class="new" title="Nhóm cộng các số nguyên modulo (trang không tồn tại)">nhóm cộng các số nguyên modulo</a> 5 <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_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3539f4e08101ff96b1587c843950789b72db0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle Z_{5}}"></span>, <i>a</i>=2 có cấp bằng 5 (vì <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 (2+2+2+2+2)\equiv 0{\pmod {5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>≡</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2+2+2+2+2)\equiv 0{\pmod {5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5853501131e0ba6db7a1ecdd12f2eb25aff12b99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.091ex; height:2.843ex;" alt="{\displaystyle (2+2+2+2+2)\equiv 0{\pmod {5}}}"></span>), nếu lấy <i>k</i>=2 thì <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^{k}=2^{2}=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{k}=2^{2}=4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/436e8647be5d8a6d7b1f6b4a7f2cbf0ce1653dd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.895ex; height:2.676ex;" alt="{\displaystyle a^{k}=2^{2}=4}"></span> sẽ có cấp bằng 2:</li></ul> <dl><dd><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 \operatorname {ord} (2^{2})={\frac {\operatorname {ord} (2)}{UCLN(\operatorname {ord} (2),2)}}={\frac {4}{2}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>U</mi> <mi>C</mi> <mi>L</mi> <mi>N</mi> <mo stretchy="false">(</mo> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} (2^{2})={\frac {\operatorname {ord} (2)}{UCLN(\operatorname {ord} (2),2)}}={\frac {4}{2}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1d9867ea53a04a2f8054681dab3dfa1f2823c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.225ex; height:6.509ex;" alt="{\displaystyle \operatorname {ord} (2^{2})={\frac {\operatorname {ord} (2)}{UCLN(\operatorname {ord} (2),2)}}={\frac {4}{2}}=2}"></span>.</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Định_lý_Cauchy_(định_lý_Côsi)"><span id=".C4.90.E1.BB.8Bnh_l.C3.BD_Cauchy_.28.C4.91.E1.BB.8Bnh_l.C3.BD_C.C3.B4si.29"></span>Định lý Cauchy (định lý Côsi)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=6" title="Sửa đổi phần “Định lý Cauchy (định lý Côsi)”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=6" title="Sửa mã nguồn tại đề mục: Định lý Cauchy (định lý Côsi)"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/%C4%90%E1%BB%8Bnh_l%C3%BD_Cauchy_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)" title="Định lý Cauchy (lý thuyết nhóm)">Định lý Cauchy</a></b>: phát biểu rằng: </p> <dl><dd>Cho nhóm <i>G</i> hữu hạn. Nếu cấp của <i>G</i> chia hết cho <i>p</i>, và <i>p</i> là số nguyên tố, thì tồn tại ít nhất một phần tử <i>a</i> thuộc <i>G</i> có cấp bằng <i>p</i>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Đồng_cấu_nhóm_và_cấp"><span id=".C4.90.E1.BB.93ng_c.E1.BA.A5u_nh.C3.B3m_v.C3.A0_c.E1.BA.A5p"></span>Đồng cấu nhóm và cấp</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=7" title="Sửa đổi phần “Đồng cấu nhóm và cấp”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=7" title="Sửa mã nguồn tại đề mục: Đồng cấu nhóm và cấp"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Cho 2 nhóm <i>G</i> và <i>H</i>, nếu <i>f</i>: <i>G</i> → <i>H</i> là một đồng cấu, <i>a</i> là một phần tử thuộc <i>G</i>, cấp của <i>a</i> là hữu hạn. Khi đó <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 \operatorname {ord} (f(a))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} (f(a))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6650b44f7417dd379b31a7886f007e7253124a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.494ex; height:2.843ex;" alt="{\displaystyle \operatorname {ord} (f(a))}"></span> là ước của <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 \operatorname {ord} (a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ord</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ord} (a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b4ca12ef9d281cb79e557706d643679bd54edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.406ex; height:2.843ex;" alt="{\displaystyle \operatorname {ord} (a)}"></span>. </p><p>Ví dụ: </p> <ul><li>Từ nhận xét trên, ta suy ra không tồn tại đồng cấu nhóm <i>h</i>: S<sub>3</sub> → <b>Z</b><sub>5</sub>, vì mọi phần tử khác 0 trong nhóm Z<sub>5</sub> đều có cấp bằng 5 và 5 không phải là ước của 1,2,3 là cấp của các phần tử trong S<sub>3</sub>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Bản_số_của_nhóm_con_chuẩn_tắc"><span id="B.E1.BA.A3n_s.E1.BB.91_c.E1.BB.A7a_nh.C3.B3m_con_chu.E1.BA.A9n_t.E1.BA.AFc"></span>Bản số của nhóm con chuẩn tắc</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=8" title="Sửa đổi phần “Bản số của nhóm con chuẩn tắc”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=8" title="Sửa mã nguồn tại đề mục: Bản số của nhóm con chuẩn tắc"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="Chú_thích"><span id="Ch.C3.BA_th.C3.ADch"></span>Chú thích</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=9" title="Sửa đổi phần “Chú thích”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=9" title="Sửa mã nguồn tại đề mục: Chú thích"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r71728118">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Nguyễn Hữu Việt Hưng, Đại số đại cương, Nhà xuất bản Giáo dục, 1999, trang 23.</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Lê Thanh Hà, Các cấu trúc đại số cơ bản, Nhà xuất bản Giáo dục, 2000, trang 41</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Tham_khảo"><span id="Tham_kh.E1.BA.A3o"></span>Tham khảo</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=10" title="Sửa đổi phần “Tham khảo”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=10" title="Sửa mã nguồn tại đề mục: Tham khảo"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading2"><h2 id="Liên_kết_ngoài"><span id="Li.C3.AAn_k.E1.BA.BFt_ngo.C3.A0i"></span>Liên kết ngoài</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&veaction=edit&section=11" title="Sửa đổi phần “Liên kết ngoài”" class="mw-editsection-visualeditor"><span>sửa</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=C%E1%BA%A5p_(l%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m)&action=edit&section=11" title="Sửa mã nguồn tại đề mục: Liên kết ngoài"><span>sửa mã nguồn</span></a><span class="mw-editsection-bracket">]</span></span></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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