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mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%97%D7%91%D7%95%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת" class=""><span>יצירת חשבון</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%97%D7%91%D7%95%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o" class=""><span>כניסה לחשבון</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="אפשרויות נוספות" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="כלים אישיים" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">כלים אישיים</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="תפריט משתמש" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_he.wikipedia.org&uselang=he"><span>תרומה לוויקיפדיה</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A8%D7%A9%D7%9E%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%97%D7%91%D7%95%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ ליצור חשבון ולהיכנס אליו, אך אין חובה לעשות זאת"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>יצירת חשבון</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%9B%D7%A0%D7%99%D7%A1%D7%94_%D7%9C%D7%97%D7%A9%D7%91%D7%95%D7%9F&returnto=%D7%97%D7%91%D7%95%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29" title="מומלץ להיכנס לחשבון, אך אין חובה לעשות זאת [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>כניסה לחשבון</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"> דפים לעורכים שלא נכנסו לחשבון <a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A8%D7%95%D7%9B%D7%99%D7%9D_%D7%94%D7%91%D7%90%D7%99%D7%9D" aria-label="מידע נוסף על עריכה"><span>מידע נוסף</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/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%AA%D7%A8%D7%95%D7%9E%D7%95%D7%AA_%D7%A9%D7%9C%D7%99" title="רשימת העריכות שנעשו מכתובת IP זו [y]" accesskey="y"><span>תרומות</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%94%D7%A9%D7%99%D7%97%D7%94_%D7%A9%D7%9C%D7%99" title="דיון על העריכות שנעשו מכתובת IP זו [n]" accesskey="n"><span>שיחה</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 id="mw-dismissablenotice-anonplace"></div><script>(function(){var node=document.getElementById("mw-dismissablenotice-anonplace");if(node){node.outerHTML="\u003Cdiv class=\"mw-dismissable-notice\"\u003E\u003Cdiv class=\"mw-dismissable-notice-close\"\u003E[\u003Ca tabindex=\"0\" role=\"button\"\u003Eהסתרה\u003C/a\u003E]\u003C/div\u003E\u003Cdiv class=\"mw-dismissable-notice-body\"\u003E\u003C!-- CentralNotice --\u003E\u003Cdiv id=\"localNotice\" data-nosnippet=\"\"\u003E\u003Cdiv class=\"anonnotice\" lang=\"he\" dir=\"rtl\"\u003E\u003Cp\u003E\u003Cb\u003Eתמיד רציתם לכתוב בוויקיפדיה אבל לא ידעתם איך? אתם מוזמנים לסדנת עריכה בוויקיפדיה. הסדנה תתקיים בספרייה הלאומית (בבניינה החדש) בירושלים ביום שישי, 06.12.24, בשעה 09:00. להרשמה לחצו \u003Ca href=\"/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:%D7%9E%D7%99%D7%96%D7%9E%D7%99_%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94/%D7%92%D7%9C%D7%90%D7%9D/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA/%D7%90%D7%99%D7%A8%D7%95%D7%A2%D7%99%D7%9D/%D7%A1%D7%93%D7%A0%D7%AA_%D7%A2%D7%A8%D7%99%D7%9B%D7%94_%D7%93%D7%A6%D7%9E%D7%91%D7%A8_2024\" title=\"ויקיפדיה:מיזמי ויקיפדיה/גלאם/הספרייה הלאומית/אירועים/סדנת עריכה דצמבר 2024\"\u003Eכאן\u003C/a\u003E.\u003C/b\u003E\n\u003C/p\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></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="אתר"> <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="תוכן עניינים" 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">תוכן עניינים</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">הסתרה</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">התחלה</div> </a> </li> <li id="toc-הגדרה" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#הגדרה"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>הגדרה</span> </div> </a> <ul id="toc-הגדרה-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-דוגמאות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#דוגמאות"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>דוגמאות</span> </div> </a> <ul id="toc-דוגמאות-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קשרים_בין_חבורות_למבנים_כלליים_יותר" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קשרים_בין_חבורות_למבנים_כלליים_יותר"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>קשרים בין חבורות למבנים כלליים יותר</span> </div> </a> <ul id="toc-קשרים_בין_חבורות_למבנים_כלליים_יותר-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-תת-חבורות" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#תת-חבורות"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>תת-חבורות</span> </div> </a> <button aria-controls="toc-תת-חבורות-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>שינוי מצב התת־פרק תת-חבורות</span> </button> <ul id="toc-תת-חבורות-sublist" class="vector-toc-list"> <li id="toc-תת-חבורות_מיוחדות" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#תת-חבורות_מיוחדות"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>תת-חבורות מיוחדות</span> </div> </a> <ul id="toc-תת-חבורות_מיוחדות-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-יוצרים_ויחסים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#יוצרים_ויחסים"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>יוצרים ויחסים</span> </div> </a> <ul id="toc-יוצרים_ויחסים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-פעולות_יסודיות_ואיברים_צמודים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#פעולות_יסודיות_ואיברים_צמודים"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>פעולות יסודיות ואיברים צמודים</span> </div> </a> <ul id="toc-פעולות_יסודיות_ואיברים_צמודים-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ראו_גם" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#ראו_גם"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>ראו גם</span> </div> </a> <ul id="toc-ראו_גם-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-קישורים_חיצוניים" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#קישורים_חיצוניים"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>קישורים חיצוניים</span> </div> </a> <ul id="toc-קישורים_חיצוניים-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="תוכן עניינים" 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="מצב תוכן העניינים" > <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">מצב תוכן העניינים</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">חבורה (מבנה אלגברי)</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="מעבר לערך בשפה אחרת. זמין ב־83 שפות" > <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-83" 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">83 שפות</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en badge-Q17437796 badge-featuredarticle mw-list-item" title="ערך מומלץ"><a href="https://en.wikipedia.org/wiki/Group_(mathematics)" title="Group (mathematics) – אנגלית" lang="en" hreflang="en" data-title="Group (mathematics)" data-language-autonym="English" data-language-local-name="אנגלית" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Groep_(wiskunde)" title="Groep (wiskunde) – אפריקאנס" lang="af" hreflang="af" data-title="Groep (wiskunde)" data-language-autonym="Afrikaans" data-language-local-name="אפריקאנס" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B2%D9%85%D8%B1%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="زمرة (رياضيات) – ערבית" lang="ar" hreflang="ar" data-title="زمرة (رياضيات)" data-language-autonym="العربية" data-language-local-name="ערבית" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D3%A9%D1%80%D0%BA%D3%A9%D0%BC_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Төркөм (математика) – בשקירית" lang="ba" hreflang="ba" data-title="Төркөм (математика)" data-language-autonym="Башҡортса" data-language-local-name="בשקירית" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Група (алгебра) – בלארוסית" lang="be" hreflang="be" data-title="Група (алгебра)" data-language-autonym="Беларуская" data-language-local-name="בלארוסית" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Група (алгебра) – בולגרית" lang="bg" hreflang="bg" data-title="Група (алгебра)" data-language-autonym="Български" data-language-local-name="בולגרית" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%97%E0%A7%8D%E0%A6%B0%E0%A7%81%E0%A6%AA_(%E0%A6%97%E0%A6%A3%E0%A6%BF%E0%A6%A4)" title="গ্রুপ (গণিত) – בנגלית" lang="bn" hreflang="bn" data-title="গ্রুপ (গণিত)" data-language-autonym="বাংলা" data-language-local-name="בנגלית" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_(matem%C3%A0tiques)" title="Grup (matemàtiques) – קטלאנית" lang="ca" hreflang="ca" data-title="Grup (matemàtiques)" data-language-autonym="Català" data-language-local-name="קטלאנית" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%88%D9%BE_(%D9%85%D8%A7%D8%AA%D9%85%D8%A7%D8%AA%DB%8C%DA%A9)" title="گرووپ (ماتماتیک) – כורדית סוראנית" lang="ckb" hreflang="ckb" data-title="گرووپ (ماتماتیک)" data-language-autonym="کوردی" data-language-local-name="כורדית סוראנית" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs badge-Q17437798 badge-goodarticle mw-list-item" title="ערך טוב"><a href="https://cs.wikipedia.org/wiki/Grupa" title="Grupa – צ׳כית" lang="cs" hreflang="cs" data-title="Grupa" data-language-autonym="Čeština" data-language-local-name="צ׳כית" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D1%88%D0%BA%C4%83%D0%BD_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ушкăн (математика) – צ׳ובשית" lang="cv" hreflang="cv" data-title="Ушкăн (математика)" data-language-autonym="Чӑвашла" data-language-local-name="צ׳ובשית" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gr%C5%B5p_(mathemateg)" title="Grŵp (mathemateg) – ולשית" lang="cy" hreflang="cy" data-title="Grŵp (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="ולשית" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Gruppe_(matematik)" title="Gruppe (matematik) – דנית" lang="da" hreflang="da" data-title="Gruppe (matematik)" data-language-autonym="Dansk" data-language-local-name="דנית" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Gruppe_(Mathematik)" title="Gruppe (Mathematik) – גרמנית" lang="de" hreflang="de" data-title="Gruppe (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="גרמנית" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9F%CE%BC%CE%AC%CE%B4%CE%B1" title="Ομάδα – יוונית" lang="el" hreflang="el" data-title="Ομάδα" data-language-autonym="Ελληνικά" data-language-local-name="יוונית" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grupo_(algebro)" title="Grupo (algebro) – אספרנטו" lang="eo" hreflang="eo" data-title="Grupo (algebro)" data-language-autonym="Esperanto" data-language-local-name="אספרנטו" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_(matem%C3%A1tica)" title="Grupo (matemática) – ספרדית" lang="es" hreflang="es" data-title="Grupo (matemática)" data-language-autonym="Español" data-language-local-name="ספרדית" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/R%C3%BChm_(matemaatika)" title="Rühm (matemaatika) – אסטונית" lang="et" hreflang="et" data-title="Rühm (matemaatika)" data-language-autonym="Eesti" data-language-local-name="אסטונית" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde_(matematika)" title="Talde (matematika) – בסקית" lang="eu" hreflang="eu" data-title="Talde (matematika)" data-language-autonym="Euskara" data-language-local-name="בסקית" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="گروه (ریاضیات) – פרסית" lang="fa" hreflang="fa" data-title="گروه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="פרסית" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ryhm%C3%A4_(algebra)" title="Ryhmä (algebra) – פינית" lang="fi" hreflang="fi" data-title="Ryhmä (algebra)" data-language-autonym="Suomi" data-language-local-name="פינית" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques) – צרפתית" lang="fr" hreflang="fr" data-title="Groupe (mathématiques)" data-language-autonym="Français" data-language-local-name="צרפתית" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Sk%C3%B6%C3%B6l_(Matematiik)" title="Skööl (Matematiik) – פריזית צפונית" lang="frr" hreflang="frr" data-title="Skööl (Matematiik)" data-language-autonym="Nordfriisk" data-language-local-name="פריזית צפונית" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Gr%C3%BApa_(matamaitic)" title="Grúpa (matamaitic) – אירית" lang="ga" hreflang="ga" data-title="Grúpa (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="אירית" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Grupo_(matem%C3%A1ticas)" title="Grupo (matemáticas) – גליסית" lang="gl" hreflang="gl" data-title="Grupo (matemáticas)" data-language-autonym="Galego" data-language-local-name="גליסית" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%AE%E0%A5%82%E0%A4%B9_(%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A4%B6%E0%A4%BE%E0%A4%B8%E0%A5%8D%E0%A4%A4%E0%A5%8D%E0%A4%B0)" title="समूह (गणितशास्त्र) – הינדי" lang="hi" hreflang="hi" data-title="समूह (गणितशास्त्र)" data-language-autonym="हिन्दी" data-language-local-name="הינדי" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) – קרואטית" lang="hr" hreflang="hr" data-title="Grupa (matematika)" data-language-autonym="Hrvatski" data-language-local-name="קרואטית" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Csoport_(matematika)" title="Csoport (matematika) – הונגרית" lang="hu" hreflang="hu" data-title="Csoport (matematika)" data-language-autonym="Magyar" data-language-local-name="הונגרית" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BD%D5%B8%D6%82%D5%B4%D5%A2_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Խումբ (մաթեմատիկա) – ארמנית" lang="hy" hreflang="hy" data-title="Խումբ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="ארמנית" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Gruppo_(mathematica)" title="Gruppo (mathematica) – ‏אינטרלינגואה" lang="ia" hreflang="ia" data-title="Gruppo (mathematica)" data-language-autonym="Interlingua" data-language-local-name="‏אינטרלינגואה" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Grup_(matematika)" title="Grup (matematika) – אינדונזית" lang="id" hreflang="id" data-title="Grup (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="אינדונזית" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Gr%C3%BApa" title="Grúpa – איסלנדית" lang="is" hreflang="is" data-title="Grúpa" data-language-autonym="Íslenska" data-language-local-name="איסלנדית" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Gruppo_(matematica)" title="Gruppo (matematica) – איטלקית" lang="it" hreflang="it" data-title="Gruppo (matematica)" data-language-autonym="Italiano" data-language-local-name="איטלקית" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%BE%A4_(%E6%95%B0%E5%AD%A6)" title="群 (数学) – יפנית" lang="ja" hreflang="ja" data-title="群 (数学)" data-language-autonym="日本語" data-language-local-name="יפנית" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AF%E1%83%92%E1%83%A3%E1%83%A4%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="ჯგუფი (მათემატიკა) – גאורגית" lang="ka" hreflang="ka" data-title="ჯგუფი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="גאורגית" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kab mw-list-item"><a href="https://kab.wikipedia.org/wiki/Tagrumma_(tusnakt)" title="Tagrumma (tusnakt) – קבילה" lang="kab" hreflang="kab" data-title="Tagrumma (tusnakt)" data-language-autonym="Taqbaylit" data-language-local-name="קבילה" class="interlanguage-link-target"><span>Taqbaylit</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Топ (математика) – קזחית" lang="kk" hreflang="kk" data-title="Топ (математика)" data-language-autonym="Қазақша" data-language-local-name="קזחית" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%97%E0%B3%8D%E0%B2%B0%E0%B3%82%E0%B2%AA%E0%B3%8D" title="ಗ್ರೂಪ್ – קנאדה" lang="kn" hreflang="kn" data-title="ಗ್ರೂಪ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="קנאדה" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0_(%EC%88%98%ED%95%99)" title="군 (수학) – קוריאנית" lang="ko" hreflang="ko" data-title="군 (수학)" data-language-autonym="한국어" data-language-local-name="קוריאנית" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Caterva_(mathematica)" title="Caterva (mathematica) – לטינית" lang="la" hreflang="la" data-title="Caterva (mathematica)" data-language-autonym="Latina" data-language-local-name="לטינית" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Grupp_(Algeber)" title="Grupp (Algeber) – לוקסמבורגית" lang="lb" hreflang="lb" data-title="Grupp (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="לוקסמבורגית" class="interlanguage-link-target"><span>Lëtzebuergesch</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Grupp_(matem%C3%A0tica)" title="Grupp (matemàtica) – לומברדית" lang="lmo" hreflang="lmo" data-title="Grupp (matemàtica)" data-language-autonym="Lombard" data-language-local-name="לומברדית" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Grup%C4%97_(algebra)" title="Grupė (algebra) – ליטאית" lang="lt" hreflang="lt" data-title="Grupė (algebra)" data-language-autonym="Lietuvių" data-language-local-name="ליטאית" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Grupa_(matem%C4%81tika)" title="Grupa (matemātika) – לטבית" lang="lv" hreflang="lv" data-title="Grupa (matemātika)" data-language-autonym="Latviešu" data-language-local-name="לטבית" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Vory_(matematika)" title="Vory (matematika) – מלגשית" lang="mg" hreflang="mg" data-title="Vory (matematika)" data-language-autonym="Malagasy" data-language-local-name="מלגשית" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%97%E0%B5%8D%E0%B4%B0%E0%B5%82%E0%B4%AA%E0%B5%8D%E0%B4%AA%E0%B5%8D" title="ഗ്രൂപ്പ് – מליאלאם" lang="ml" hreflang="ml" data-title="ഗ്രൂപ്പ്" data-language-autonym="മലയാളം" data-language-local-name="מליאלאם" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kumpulan_(matematik)" title="Kumpulan (matematik) – מלאית" lang="ms" hreflang="ms" data-title="Kumpulan (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="מלאית" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mt mw-list-item"><a href="https://mt.wikipedia.org/wiki/Grupp_(matematika)" title="Grupp (matematika) – מלטית" lang="mt" hreflang="mt" data-title="Grupp (matematika)" data-language-autonym="Malti" data-language-local-name="מלטית" class="interlanguage-link-target"><span>Malti</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Groep_(wiskunde)" title="Groep (wiskunde) – הולנדית" lang="nl" hreflang="nl" data-title="Groep (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="הולנדית" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_gruppe" title="Matematisk gruppe – נורווגית חדשה" lang="nn" hreflang="nn" data-title="Matematisk gruppe" data-language-autonym="Norsk nynorsk" data-language-local-name="נורווגית חדשה" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Gruppe_(matematikk)" title="Gruppe (matematikk) – נורווגית ספרותית" lang="nb" hreflang="nb" data-title="Gruppe (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="נורווגית ספרותית" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Grupe_(matematike)" title="Grupe (matematike) – נוביאל" lang="nov" hreflang="nov" data-title="Grupe (matematike)" data-language-autonym="Novial" data-language-local-name="נוביאל" class="interlanguage-link-target"><span>Novial</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Grop_(matematicas)" title="Grop (matematicas) – אוקסיטנית" lang="oc" hreflang="oc" data-title="Grop (matematicas)" data-language-autonym="Occitan" data-language-local-name="אוקסיטנית" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Grupa_(matematyka)" title="Grupa (matematyka) – פולנית" lang="pl" hreflang="pl" data-title="Grupa (matematyka)" data-language-autonym="Polski" data-language-local-name="פולנית" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Strop" title="Strop – פיימונטית" lang="pms" hreflang="pms" data-title="Strop" data-language-autonym="Piemontèis" data-language-local-name="פיימונטית" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="گروہ (ریاضی) – פנג׳בית מערבית" lang="pnb" hreflang="pnb" data-title="گروہ (ریاضی)" data-language-autonym="پنجابی" data-language-local-name="פנג׳בית מערבית" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_(matem%C3%A1tica)" title="Grupo (matemática) – פורטוגזית" lang="pt" hreflang="pt" data-title="Grupo (matemática)" data-language-autonym="Português" data-language-local-name="פורטוגזית" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro badge-Q17437796 badge-featuredarticle mw-list-item" title="ערך מומלץ"><a href="https://ro.wikipedia.org/wiki/Grup_(matematic%C4%83)" title="Grup (matematică) – רומנית" lang="ro" hreflang="ro" data-title="Grup (matematică)" data-language-autonym="Română" data-language-local-name="רומנית" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Группа (математика) – רוסית" lang="ru" hreflang="ru" data-title="Группа (математика)" data-language-autonym="Русский" data-language-local-name="רוסית" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Gruppu_(matimatica)" title="Gruppu (matimatica) – סיציליאנית" lang="scn" hreflang="scn" data-title="Gruppu (matimatica)" data-language-autonym="Sicilianu" data-language-local-name="סיציליאנית" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) – סרבו-קרואטית" lang="sh" hreflang="sh" data-title="Grupa (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="סרבו-קרואטית" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Group_(mathematics)" title="Group (mathematics) – אנגלית פשוטה" lang="en-simple" hreflang="en-simple" data-title="Group (mathematics)" data-language-autonym="Simple English" data-language-local-name="אנגלית פשוטה" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Grupa_(matematika)" title="Grupa (matematika) – סלובקית" lang="sk" hreflang="sk" data-title="Grupa (matematika)" data-language-autonym="Slovenčina" data-language-local-name="סלובקית" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Grupa" title="Grupa – סלובנית" lang="sl" hreflang="sl" data-title="Grupa" data-language-autonym="Slovenščina" data-language-local-name="סלובנית" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Група (математика) – סרבית" lang="sr" hreflang="sr" data-title="Група (математика)" data-language-autonym="Српски / srpski" data-language-local-name="סרבית" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Grupp_(matematik)" title="Grupp (matematik) – שוודית" lang="sv" hreflang="sv" data-title="Grupp (matematik)" data-language-autonym="Svenska" data-language-local-name="שוודית" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-szl mw-list-item"><a href="https://szl.wikipedia.org/wiki/Grupa_(matymatyka)" title="Grupa (matymatyka) – שלזית" lang="szl" hreflang="szl" data-title="Grupa (matymatyka)" data-language-autonym="Ślůnski" data-language-local-name="שלזית" class="interlanguage-link-target"><span>Ślůnski</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%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="குலம் (கணிதம்) – טמילית" lang="ta" hreflang="ta" data-title="குலம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="טמילית" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%93%D1%83%D1%80%D3%AF%D2%B3_(%D1%80%D0%B8%D1%91%D0%B7%D0%B8%D1%91%D1%82)" title="Гурӯҳ (риёзиёт) – טג׳יקית" lang="tg" hreflang="tg" data-title="Гурӯҳ (риёзиёт)" data-language-autonym="Тоҷикӣ" data-language-local-name="טג׳יקית" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%A3%E0%B8%B8%E0%B8%9B_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="กรุป (คณิตศาสตร์) – תאית" lang="th" hreflang="th" data-title="กรุป (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="תאית" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Grupo_(matematika)" title="Grupo (matematika) – טאגאלוג" lang="tl" hreflang="tl" data-title="Grupo (matematika)" data-language-autonym="Tagalog" data-language-local-name="טאגאלוג" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Grup_(matematik)" title="Grup (matematik) – טורקית" lang="tr" hreflang="tr" data-title="Grup (matematik)" data-language-autonym="Türkçe" data-language-local-name="טורקית" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%93%D1%80%D1%83%D0%BF%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Група (математика) – אוקראינית" lang="uk" hreflang="uk" data-title="Група (математика)" data-language-autonym="Українська" data-language-local-name="אוקראינית" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%AF%D8%B1%D9%88%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="گروہ (ریاضی) – אורדו" lang="ur" hreflang="ur" data-title="گروہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="אורדו" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi badge-Q17437796 badge-featuredarticle mw-list-item" title="ערך מומלץ"><a href="https://vi.wikipedia.org/wiki/Nh%C3%B3m_(to%C3%A1n_h%E1%BB%8Dc)" title="Nhóm (toán học) – וייטנאמית" lang="vi" hreflang="vi" data-title="Nhóm (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="וייטנאמית" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-vls mw-list-item"><a href="https://vls.wikipedia.org/wiki/Groep_(algebra)" title="Groep (algebra) – פלמית מערבית" lang="vls" hreflang="vls" data-title="Groep (algebra)" data-language-autonym="West-Vlams" data-language-local-name="פלמית מערבית" class="interlanguage-link-target"><span>West-Vlams</span></a></li><li class="interlanguage-link interwiki-vo mw-list-item"><a href="https://vo.wikipedia.org/wiki/Grup" title="Grup – ‏וולאפיק" lang="vo" hreflang="vo" data-title="Grup" data-language-autonym="Volapük" data-language-local-name="‏וולאפיק" class="interlanguage-link-target"><span>Volapük</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BE%A4" title="群 – סינית וו" lang="wuu" hreflang="wuu" data-title="群" data-language-autonym="吴语" data-language-local-name="סינית וו" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%92%D7%A8%D7%95%D7%A4%D7%A2_(%D7%9E%D7%90%D7%98%D7%A2%D7%9E%D7%90%D7%98%D7%99%D7%A7)" title="גרופע (מאטעמאטיק) – יידיש" lang="yi" hreflang="yi" data-title="גרופע (מאטעמאטיק)" data-language-autonym="ייִדיש" data-language-local-name="יידיש" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%BE%A4" title="群 – סינית" lang="zh" hreflang="zh" data-title="群" data-language-autonym="中文" data-language-local-name="סינית" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%BE%A4_(%E4%BB%A3%E6%95%B8)" title="群 (代數) – סינית קלאסית" lang="lzh" hreflang="lzh" data-title="群 (代數)" data-language-autonym="文言" data-language-local-name="סינית קלאסית" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/K%C3%BBn_(s%C3%B2%CD%98-ha%CC%8Dk)" title="Kûn (sò͘-ha̍k) – מין נאנית" lang="nan" hreflang="nan" data-title="Kûn (sò͘-ha̍k)" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="מין נאנית" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%BE%A3" title="羣 – קנטונזית" lang="yue" hreflang="yue" data-title="羣" data-language-autonym="粵語" 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class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:%D7%A1%D7%A4%D7%A8&bookcmd=book_creator&referer=%D7%97%D7%91%D7%95%D7%A8%D7%94+%28%D7%9E%D7%91%D7%A0%D7%94+%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29"><span>יצירת ספר</span></a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D7%9E%D7%99%D7%95%D7%97%D7%93:DownloadAsPdf&page=%D7%97%D7%91%D7%95%D7%A8%D7%94_%28%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%29&action=show-download-screen"><span>הורדה כ־PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&printable=yes" title="גרסה להדפסה של הדף הזה [p]" accesskey="p"><span>גרסה להדפסה</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> במיזמים אחרים </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Group_theory" hreflang="en"><span>ויקישיתוף</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q83478" title="קישור לפריט המשויך במאגר הנתונים [g]" accesskey="g"><span>פריט ויקינתונים</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="כלי דף"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="מראה"> <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">מראה</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">העברה לסרגל הצד</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">הסתרה</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">מתוך ויקיפדיה, האנציקלופדיה החופשית</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(הופנה מהדף <a href="/w/index.php?title=%D7%AA%D7%AA-%D7%97%D7%91%D7%95%D7%A8%D7%94&redirect=no" class="mw-redirect" title="תת-חבורה">תת-חבורה</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-rtl mw-parser-output" lang="he" dir="rtl"><table cellpadding="1" style="float: left; clear:both; border: 1px solid #8888aa; background: #f7f8ff; padding: 5px; font-size: 95%; margin: 0.5em 1em 0.5em 0.5em;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/60px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/90px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/120px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> <p>בערך זה<br />נעשה שימוש<br />בסימנים מוסכמים<br />מתחום המתמטיקה.<br />להבהרת הסימנים<br />ראו <a href="/wiki/%D7%A1%D7%99%D7%9E%D7%95%D7%9F_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="סימון מתמטי">סימון מתמטי</a>. </p> </td></tr></tbody></table> <p>ב<a href="/wiki/%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="מתמטיקה">מתמטיקה</a>, <b>חבורה</b> (Group) היא <a href="/wiki/%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מבנה אלגברי">מבנה אלגברי</a> המורכב מ<a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קבוצה (מתמטיקה)">קבוצה</a> ו<a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%AA" title="פעולה בינארית">פעולה בינארית</a> <a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה אסוציאטיבית">אסוציאטיבית</a> (קיבוצית). </p><p>החבורות הופיעו במחקר המתמטי במהלך <a href="/wiki/%D7%94%D7%9E%D7%90%D7%94_%D7%94-19" title="המאה ה-19">המאה ה-19</a>, במסגרת הניסיונות לפתור משוואות <a href="/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9D" title="פולינום">פולינומיות</a> ממעלה גבוהה, כדוגמת הפתרונות ל<a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%9E%D7%9E%D7%A2%D7%9C%D7%94_%D7%A9%D7%9C%D7%99%D7%A9%D7%99%D7%AA" title="משוואה ממעלה שלישית">משוואה ממעלה שלישית</a> ו<a href="/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%94_%D7%9E%D7%9E%D7%A2%D7%9C%D7%94_%D7%A8%D7%91%D7%99%D7%A2%D7%99%D7%AA" title="משוואה ממעלה רביעית">רביעית</a> שהתגלו ב<a href="/wiki/%D7%94%D7%9E%D7%90%D7%94_%D7%94-16" title="המאה ה-16">מאה ה-16</a>. החבורות שבהן עסקו החוקרים הראשונים, ובראשם <a href="/wiki/%D7%90%D7%95%D7%95%D7%A8%D7%99%D7%A1%D7%98_%D7%92%D7%9C%D7%95%D7%90%D7%94" title="אווריסט גלואה">גלואה</a>, היו חבורות ספציפיות שאיבריהן הם <a href="/wiki/%D7%AA%D7%9E%D7%95%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="תמורה (מתמטיקה)">תמורות</a>. מאוחר יותר ניסח <a href="/wiki/%D7%90%D7%A8%D7%AA%D7%95%D7%A8_%D7%A7%D7%99%D7%99%D7%9C%D7%99" title="ארתור קיילי">ארתור קיילי</a> את מערכת ה<a href="/wiki/%D7%90%D7%A7%D7%A1%D7%99%D7%95%D7%9E%D7%94" title="אקסיומה">אקסיומות</a> המגדירה חבורה באופן מופשט, וייסד בכך את <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA" title="תורת החבורות">תורת החבורות</a>. </p><p>לתורת החבורות יש שימושים רבים במתמטיקה עצמה, כאמור, אך גם ב<a href="/wiki/%D7%A4%D7%99%D7%96%D7%99%D7%A7%D7%94" title="פיזיקה">פיזיקה</a>, כמו בחקר מבנה ה<a href="/wiki/%D7%92%D7%91%D7%99%D7%A9" title="גביש">גבישים</a> וה<a href="/wiki/%D7%9E%D7%95%D7%9C%D7%A7%D7%95%D7%9C%D7%94" title="מולקולה">מולקולות</a>, ובחקר מושג ה<a href="/wiki/%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%94" title="סימטריה">סימטריה</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="הגדרה"><span id=".D7.94.D7.92.D7.93.D7.A8.D7.94"></span>הגדרה</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=1" title="עריכת קוד המקור של הפרק: הגדרה"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=1" title="עריכת פסקה: "הגדרה"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>חבורה</b> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> היא <a href="/wiki/%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מבנה אלגברי">מבנה אלגברי</a> בסיסי הכולל <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="קבוצה (מתמטיקה)">קבוצה</a> עם <a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%91%D7%99%D7%A0%D7%90%D7%A8%D7%99%D7%AA" title="פעולה בינארית">פעולה בינארית</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 *}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∗</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle *}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9972f426d9e07855984f73ee195a21dbc21755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}"></span> ("<a href="/wiki/%D7%A1%D7%92%D7%99%D7%A8%D7%95%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="סגירות (אלגברה)">סגורה</a>": לכל <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,b\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b3751cbf424027e1e00e22d191a2d465403c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.929ex; height:2.509ex;" alt="{\displaystyle a,b\in G}"></span> מתקיים כי <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*b\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*b\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb72f6dc15c7bf9ad764645b597e1223819611d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.089ex; height:2.176ex;" alt="{\displaystyle a*b\in G}"></span>), אשר מקיימת את התכונות הבאות: </p> <ul><li><a href="/wiki/%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%95%D7%AA" class="mw-redirect" title="אסוציאטיביות">אסוציאטיביות</a> (קיבוציות): לכל <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,b,c\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b,c\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11960c7b14a27f9720d2011cd9116f1bdd22fecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.969ex; height:2.509ex;" alt="{\displaystyle a,b,c\in G}"></span> מתקיים <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*(b*c)=(a*b)*c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>∗</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*(b*c)=(a*b)*c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb031018f88b92239cb9d9a06e2ad32ce39b258c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.964ex; height:2.843ex;" alt="{\displaystyle a*(b*c)=(a*b)*c}"></span>.</li> <li>קיום <a href="/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="איבר יחידה">איבר יחידה</a>: קיים איבר <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 e\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc993b87bf6f6f7f8e5f1e99011f92bd4b0188a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.751ex; height:2.176ex;" alt="{\displaystyle e\in G}"></span> כך שלכל <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\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9f5ea7aea0b7a62b07eae139e7a5038ea5a120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle a\in G}"></span> מתקיים <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*e=e*a=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>e</mi> <mo>=</mo> <mi>e</mi> <mo>∗</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*e=e*a=a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d83cadf2ea5c5fab2357795b7ec9495173bd3099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.443ex; height:1.676ex;" alt="{\displaystyle a*e=e*a=a}"></span>.</li> <li>כל האיברים <a href="/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%94%D7%A4%D7%99%D7%9A" title="איבר הפיך">הפיכים</a>: לכל <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\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe9f5ea7aea0b7a62b07eae139e7a5038ea5a120" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.897ex; height:2.176ex;" alt="{\displaystyle a\in G}"></span> קיים <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 b\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da5cc273752cfc08ad479f4f3397110b48b178d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.665ex; height:2.176ex;" alt="{\displaystyle b\in G}"></span> עבורו <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*b=b*a=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>∗</mo> <mi>a</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*b=b*a=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2273a34a8cc00d7b41dc47fef8889c38e0cbc0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.125ex; height:2.176ex;" alt="{\displaystyle a*b=b*a=e}"></span>.</li></ul> <p>(מהאקסיומות נובע שיש רק איבר יחידה אחד, ושלכל איבר יש הופכי אחד; די להניח שכל איבר הפיך משמאל). </p><p><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%90%D7%91%D7%9C%D7%99%D7%AA" title="חבורה אבלית">חבורה אבלית</a> (חילופית) היא חבורה שבה מתקיים, בנוסף, תנאי ה<a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%A7%D7%95%D7%9E%D7%95%D7%98%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה קומוטטיבית">קומוטטיביות</a> (חילופיות) <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*b=b*a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∗</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>∗</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a*b=b*a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7a1235743322cc522c4ecddbc3bb6ca354eb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.943ex; height:2.176ex;" alt="{\displaystyle a*b=b*a}"></span> לכל <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,b\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4b3751cbf424027e1e00e22d191a2d465403c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.929ex; height:2.509ex;" alt="{\displaystyle a,b\in G}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="דוגמאות"><span id=".D7.93.D7.95.D7.92.D7.9E.D7.90.D7.95.D7.AA"></span>דוגמאות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=2" title="עריכת קוד המקור של הפרק: דוגמאות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=2" title="עריכת פסקה: "דוגמאות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D7%97%D7%95%D7%92_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%A9%D7%9C%D7%9E%D7%99%D7%9D" title="חוג המספרים השלמים">חוג המספרים השלמים</a> עם פעולת החיבור הוא חבורה אבלית הנקראת <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A6%D7%99%D7%A7%D7%9C%D7%99%D7%AA" title="חבורה ציקלית">החבורה הציקלית האינסופית</a>. זו אינה חבורה ביחס לכפל, משום שיש מספרים שאין להם הפכי שלם.</li> <li><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%95%D7%AA" title="חבורת סימטריות">קבוצת הסימטריות</a> של <a href="/wiki/%D7%9E%D7%A6%D7%95%D7%9C%D7%A2_%D7%9E%D7%A9%D7%95%D7%9B%D7%9C%D7%9C" title="מצולע משוכלל">מצולע משוכלל</a> (פעולות שיקוף וסיבוב שלא משנות אותו) עם פעולת ה<a href="/wiki/%D7%94%D7%A8%D7%9B%D7%91%D7%AA_%D7%A4%D7%95%D7%A0%D7%A7%D7%A6%D7%99%D7%95%D7%AA" title="הרכבת פונקציות">הרכבה</a> (ביצוע הפעולות בזה אחרי זה). חבורה זו נקראת <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%93%D7%99%D7%94%D7%93%D7%A8%D7%9C%D7%99%D7%AA" title="חבורה דיהדרלית">החבורה הדיהדרלית</a>.</li> <li><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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f049ac28d4ac8097b625f9d71c1f22b2ebd1bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.643ex; height:2.509ex;" alt="{\displaystyle S_{n}}"></span>, <a href="/wiki/%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%94%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="החבורה הסימטרית">חבורת התמורות</a> על <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> איברים, ביחס לפעולת ההרכבה. זוהי חבורה לא אבלית לכל <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 n>2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}"></span>.</li> <li>קבוצת ה<a href="/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%94%D7%A4%D7%99%D7%9B%D7%94" title="מטריצה הפיכה">מטריצות ההפיכות</a> מסדר <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> מעל שדה <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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span>, המסומנת <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 {\text{GL}}_{n}{\text{(F)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>GL</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>(F)</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{GL}}_{n}{\text{(F)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf53854cb6e894329ff75b6c07fcaf4f344040fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.823ex; height:2.843ex;" alt="{\displaystyle {\text{GL}}_{n}{\text{(F)}}}"></span>, היא חבורה ביחס לפעולת <a href="/wiki/%D7%9B%D7%A4%D7%9C_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="כפל מטריצות">כפל מטריצות</a>. חבורה זו נקראת <a href="/wiki/%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%94%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA_%D7%94%D7%9B%D7%9C%D7%9C%D7%99%D7%AA" title="החבורה הליניארית הכללית">החבורה הליניארית הכללית</a>.</li> <li>חבורת <a href="/wiki/%D7%A9%D7%95%D7%A8%D7%A9%D7%99_%D7%99%D7%97%D7%99%D7%93%D7%94" class="mw-redirect" title="שורשי יחידה">שורשי היחידה</a> מסדר <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, כלומר השורשים המרוכבים של המשוואה <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^{n}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{n}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34fb31edf665701c275c44a5be8b82a95509888d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.57ex; height:2.343ex;" alt="{\displaystyle z^{n}=1}"></span> עם פעולת ה<a href="/wiki/%D7%A9%D7%93%D7%94_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%9E%D7%A8%D7%95%D7%9B%D7%91%D7%99%D7%9D" title="שדה המספרים המרוכבים">כפל של מספרים מרוכבים</a>. חבורה זו היא <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A6%D7%99%D7%A7%D7%9C%D7%99%D7%AA" title="חבורה ציקלית">חבורה ציקלית</a> מ<a href="/wiki/%D7%A1%D7%93%D7%A8_%D7%A9%D7%9C_%D7%97%D7%91%D7%95%D7%A8%D7%94" class="mw-redirect" title="סדר של חבורה">סדר</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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>. גם <a href="/wiki/%D7%9E%D7%A2%D7%92%D7%9C_%D7%94%D7%99%D7%97%D7%99%D7%93%D7%94" title="מעגל היחידה">מעגל היחידה</a> ב-<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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>, כלומר <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\in \mathbb {C} :|z|=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{z\in \mathbb {C} :|z|=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e13458dc89d403bd8a59644a1d92d4be35c613d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.512ex; height:2.843ex;" alt="{\displaystyle \{z\in \mathbb {C} :|z|=1\}}"></span>, הוא חבורה ביחס לכפל.</li></ul> <div class="mw-heading mw-heading2"><h2 id="קשרים_בין_חבורות_למבנים_כלליים_יותר"><span id=".D7.A7.D7.A9.D7.A8.D7.99.D7.9D_.D7.91.D7.99.D7.9F_.D7.97.D7.91.D7.95.D7.A8.D7.95.D7.AA_.D7.9C.D7.9E.D7.91.D7.A0.D7.99.D7.9D_.D7.9B.D7.9C.D7.9C.D7.99.D7.99.D7.9D_.D7.99.D7.95.D7.AA.D7.A8"></span>קשרים בין חבורות למבנים כלליים יותר</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=3" title="עריכת קוד המקור של הפרק: קשרים בין חבורות למבנים כלליים יותר"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=3" title="עריכת פסקה: "קשרים בין חבורות למבנים כלליים יותר"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" style="text-align: center;"> <caption>מבנים אלגבריים (תחום החבורות) </caption> <tbody><tr> <th>שם </th> <th><a href="/wiki/%D7%A1%D7%92%D7%99%D7%A8%D7%95%D7%AA_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="סגירות (אלגברה)">סגירות</a> </th> <th><a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה אסוציאטיבית">אסוציאטיביות</a> </th> <th><a href="/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%99%D7%97%D7%99%D7%93%D7%94" title="איבר יחידה">איבר יחידה</a> </th> <th><a href="/wiki/%D7%90%D7%99%D7%91%D7%A8_%D7%94%D7%95%D7%A4%D7%9B%D7%99" title="איבר הופכי">איבר הופכי</a> </th> <th><a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%94_%D7%A7%D7%95%D7%9E%D7%95%D7%98%D7%98%D7%99%D7%91%D7%99%D7%AA" title="פעולה קומוטטיבית">קומוטטביות</a> </th></tr> <tr> <th><a href="/wiki/%D7%9E%D7%90%D7%92%D7%9E%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מאגמה (מבנה אלגברי)">מאגמה</a> </th> <td style="background: lime"><b>כן</b></td> <td style="background: #F1948A"><b>לא</b> </td> <td style="background: #F1948A"><b>לא</b></td> <td style="background: #F1948A"><b>לא</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th><a href="/wiki/%D7%A7%D7%95%D7%95%D7%90%D7%96%D7%99-%D7%97%D7%91%D7%95%D7%A8%D7%94" title="קוואזי-חבורה">קוואזי-חבורה</a> </th> <td style="background: lime"><b>כן</b></td> <td style="background: #F1948A"><b>לא</b> </td> <td style="background: #F1948A"><b>לא</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th>לולאה </th> <td style="background: lime"><b>כן</b></td> <td style="background: #F1948A"><b>לא</b> </td> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%9C%D7%9E%D7%97%D7%A6%D7%94" title="חבורה למחצה">חבורה למחצה</a> </th> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: #F1948A"><b>לא</b></td> <td style="background: #F1948A"><b>לא</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%9C%D7%9E%D7%97%D7%A6%D7%94#חבורה_למחצה_הפיכה" title="חבורה למחצה">חבורה למחצה הפיכה</a> </th> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: #F1948A"><b>לא</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th><a href="/wiki/%D7%9E%D7%95%D7%A0%D7%95%D7%90%D7%99%D7%93_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מונואיד (מבנה אלגברי)">מונואיד</a> </th> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: lime"><b>כן</b></td> <td style="background: #F1948A"><b>לא</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th>חבורה </th> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: #F1948A"><b>לא</b> </td></tr> <tr> <th><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%90%D7%91%D7%9C%D7%99%D7%AA" title="חבורה אבלית">חבורה אבלית</a> </th> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: lime"><b>כן</b></td> <td style="background: lime"><b>כן</b> </td> <td style="background: lime"><b>כן</b> </td></tr> </tbody></table> <p>איבר היחידה של חבורה הוא ה<a href="/wiki/%D7%90%D7%99%D7%93%D7%9E%D7%A4%D7%95%D7%98%D7%A0%D7%98" title="אידמפוטנט">אידמפוטנט</a> היחיד בה. ב<a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%9C%D7%9E%D7%97%D7%A6%D7%94" title="חבורה למחצה">חבורה למחצה</a> יש בדרך כלל אידמפוטנטים רבים, והקשרים ביניהם מאפשרים לבנות במדורג משפחות של חבורות למחצה שיש להן דמיון מסוים לחבורות. למשל, <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%9C%D7%9E%D7%97%D7%A6%D7%94_%D7%94%D7%A4%D7%99%D7%9B%D7%94" class="mw-redirect" title="חבורה למחצה הפיכה">חבורה למחצה הפיכה</a> היא חבורה למחצה שבה לכל <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> קיים <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> יחיד עבורו <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 xyx=x,yxy=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyx=x,yxy=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db4e635713f61a6fee743056fa3f8257ad651c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.172ex; height:2.009ex;" alt="{\displaystyle xyx=x,yxy=y}"></span>; במקרה זה מסמנים <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 x^{-1}=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{-1}=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0a65b801bba5c946a77d3b524ded7e0c0083459" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.916ex; height:3.009ex;" alt="{\displaystyle x^{-1}=y}"></span>. בחבורה למחצה סופית <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span>, לכל אידמפוטנט <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span>, קבוצת האיברים המקיימים <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 xx^{-1}=x^{-1}x=e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> <mo>=</mo> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xx^{-1}=x^{-1}x=e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/908561b1ed1f43636e88ca22c2808269f2fa5899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.265ex; height:2.676ex;" alt="{\displaystyle xx^{-1}=x^{-1}x=e}"></span> היא תת-החבורה המקסימלית של <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> אשר <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 e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"></span> הוא איבר היחידה שלה. </p><p>ב<a href="/wiki/%D7%9E%D7%95%D7%A0%D7%95%D7%90%D7%99%D7%93" class="mw-redirect" title="מונואיד">מונואיד</a>, שהוא חבורה למחצה עם איבר יחידה 1, אפשר לדרוש שאיבר <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> יהיה "הפיך מימין" (קיים <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> עבורו <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704b4c91a503eed5a4618aab78777fb1c354ab75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.488ex; height:2.176ex;" alt="{\displaystyle ab=1}"></span>) או "הפיך משמאל" (קיים <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> עבורו <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 ba=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ba=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2776b967fc31a08e91623fda454a6c74a888147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.488ex; height:2.176ex;" alt="{\displaystyle ba=1}"></span>). איבר <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> שהוא הפיך גם מימין וגם משמאל הוא הפיך, כלומר, קיים <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> המקיים בו-זמנית <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=ba=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=ba=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e129af82154ac2388aa6621fda5d0e008b82c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.814ex; height:2.176ex;" alt="{\displaystyle ab=ba=1}"></span>. בכל מונואיד, אוסף האיברים ההפיכים סגור לכפל (בגלל התכונה <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)^{-1}=b^{-1}a^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab)^{-1}=b^{-1}a^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f5c96926b3d8867843b4df0ccc01709d76a45f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.361ex; height:3.176ex;" alt="{\displaystyle (ab)^{-1}=b^{-1}a^{-1}}"></span>), והוא מהווה לכן חבורה. אם כל איבר של המונואיד הוא הפיך משמאל, אז כולם הפיכים, והמונואיד הוא חבורה. לעומת זאת קיימים מונואידים שאינם חבורות שבהם לכל <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> קיימים <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 x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span> עבורם <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 xay=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>a</mi> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xay=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c90357b08c0c895a9173539220d79ca07d5b6140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.976ex; height:2.509ex;" alt="{\displaystyle xay=1}"></span>. מונואיד שבו מהשוויון <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 ax=ay}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax=ay}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d50d94841ec9f3f4bd55768ccd6dda806e1bf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.043ex; height:2.009ex;" alt="{\displaystyle ax=ay}"></span> תמיד נובע <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 x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/409a91214d63eabe46ec10ff3cbba689ab687366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x=y}"></span>, נקרא "מונואיד עם צמצום משמאל"; כל מונואיד הניתן לשיכון בחבורה הוא בעל צמצום (מימין ומשמאל), אבל ההפך אינו נכון. מונואיד סופי עם צמצום משמאל הוא חבורה. </p> <div class="mw-heading mw-heading2"><h2 id="תת-חבורות"><span id=".D7.AA.D7.AA-.D7.97.D7.91.D7.95.D7.A8.D7.95.D7.AA"></span>תת-חבורות</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=4" title="עריכת קוד המקור של הפרק: תת-חבורות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=4" title="עריכת פסקה: "תת-חבורות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>תת-קבוצה של חבורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> המהווה בעצמה חבורה (ביחס לאותה פעולה בינארית אסוציאטיבית ולאותו איבר יחידה), נקראת <b>תת-חבורה</b>. כל תת-קבוצה הכוללת יחד עם כל איבר את ההפכי שלו, ויחד עם כל שני איברים את מכפלתם, היא תת-חבורה. ה<a href="/wiki/%D7%97%D7%99%D7%AA%D7%95%D7%9A_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="חיתוך (מתמטיקה)">חיתוך</a> של תת-חבורות הוא תמיד תת-חבורה. כל תת-חבורה <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> מחלקת את החבורה ל<a href="/wiki/%D7%9E%D7%97%D7%9C%D7%A7%D7%95%D7%AA_%D7%A9%D7%A7%D7%99%D7%9C%D7%95%D7%AA" class="mw-redirect" title="מחלקות שקילות">מחלקות שקילות</a>, הנקראות <a href="/wiki/%D7%A7%D7%95%D7%A1%D7%98" class="mw-redirect" title="קוסט">קוסטים</a>, בשני אופנים: מימין, המחלקות הן מהצורה <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 gH=\{gh:h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>H</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mi>h</mi> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gH=\{gh:h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52abd96e986df1982027f24fc3e5554360ac2a53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.238ex; height:2.843ex;" alt="{\displaystyle gH=\{gh:h\in H\}}"></span>, ומשמאל, המחלקות הן מהצורה <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 Hg=\{hg:h\in H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mi>g</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mi>g</mi> <mo>:</mo> <mi>h</mi> <mo>∈</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Hg=\{hg:h\in H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05943d93c582da30f3abd8dd77906aafb6e7ca1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.238ex; height:2.843ex;" alt="{\displaystyle Hg=\{hg:h\in H\}}"></span>. מספר האיברים בכל מחלקה (ימנית או שמאלית) שווה למספר האיברים בתת-החבורה, ומכאן נובע <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%9C%D7%92%D7%A8%D7%90%D7%A0%D7%96%27_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="משפט לגראנז' (תורת החבורות)">משפט לגראנז'</a>: ה<a href="/wiki/%D7%A1%D7%93%D7%A8_%D7%A9%D7%9C_%D7%97%D7%91%D7%95%D7%A8%D7%94" class="mw-redirect" title="סדר של חבורה">סדר</a> של תת-חבורה של חבורה סופית, מחלק את הסדר של החבורה. מספר המחלקות (השמאליות או הימניות) של תת-חבורה נקרא ה<b>אינדקס</b> של תת-החבורה ומסומן <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [G:H]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c19ed6f18e6db133b5a0257ecde8026808fd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.121ex; height:2.843ex;" alt="{\displaystyle [G:H]}"></span>. כאשר החבורות סופיות מתקיים <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [G:H]={\frac {|G|}{|H|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>G</mi> <mo>:</mo> <mi>H</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [G:H]={\frac {|G|}{|H|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cdff77dd1c148c730742f502f65fe0b025f92033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.413ex; height:6.509ex;" alt="{\displaystyle [G:H]={\frac {|G|}{|H|}}}"></span>. </p><p>מהסדר של חבורה (סופית) אפשר להסיק רבות על המבנה שלה. בין המשפטים הבסיסיים בכיוון זה אפשר למנות את <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%95%D7%A9%D7%99_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="משפט קושי (תורת החבורות)">משפט קושי</a> על קיומם של איברים בעלי סדר ראשוני, ואת <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%A1%D7%99%D7%9C%D7%95" title="משפטי סילו">משפטי סילו</a> על קיומן של תת-חבורות שסדרן הוא חזקה של ראשוני. </p><p>בין תת-החבורות, חשובות במיוחד תת-החבורות ה<a href="/wiki/%D7%AA%D7%AA_%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" class="mw-redirect" title="תת חבורה נורמלית">נורמליות</a>, שהן תת-חבורות הכוללות, יחד עם כל איבר, את כל האיברים הצמודים לו. במילים אחרות, תת-חבורה <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> היא תת-חבורה נורמלית של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> אם לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1be73903416a0dd94b8cbc2268ce480810c0e62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.783ex; height:2.509ex;" alt="{\displaystyle g\in G}"></span> מתקיים <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 gHg^{-1}\subset H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>H</mi> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⊂</mo> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gHg^{-1}\subset H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d650420b954766c59f59c883a3c746b47c9fc4b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.793ex; height:3.009ex;" alt="{\displaystyle gHg^{-1}\subset H}"></span>; לכן <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 gH=Hg}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mi>H</mi> <mo>=</mo> <mi>H</mi> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle gH=Hg}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d1dec58109c201c75ce8384c01a1aba2c6fcd03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.458ex; height:2.509ex;" alt="{\displaystyle gH=Hg}"></span> לכל <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>. מכאן נובע שהמחלקות הימניות והשמאליות של תת-חבורה נורמלית שוות זו לזו. על אוסף המחלקות ביחס לתת-חבורה נורמלית <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> אפשר להגדיר פעולת כפל, ההופכת אותו לחבורה; חבורה זו נקראת <b>חבורת המנה</b> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> ביחס ל-<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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, וגודלה הוא האינדקס של <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>. החבורה עצמה, ותת-החבורה הכוללת את איבר היחידה בלבד, הן תמיד נורמליות. חבורה שאין לה תת-חבורות נורמליות אחרות נקראת <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A4%D7%A9%D7%95%D7%98%D7%94" title="חבורה פשוטה">חבורה פשוטה</a>. חבורה שאינה פשוטה אפשר לבנות מתת-חבורה נורמלית ומחבורת המנה, באמצעות תהליך הנקרא <a href="/w/index.php?title=%D7%94%D7%A8%D7%97%D7%91%D7%94_%D7%A9%D7%9C_%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA&action=edit&redlink=1" class="new" title="הרחבה של חבורות (הדף אינו קיים)">הרחבה של חבורות</a>. נורמליות אינה עוברת בתורשה (כלומר, ייתכן ש-<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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> תת-חבורה נורמלית של <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, ו-<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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> תת-חבורה נורמלית של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, בעוד כי <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> אינה נורמלית ב-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>). </p><p>ה<b>מכפלה</b> של תת-חבורות מוגדרת לפי הכפלת האיברים, <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 H_{1}H_{2}=\{h_{1}h_{2}:h_{1}\in H_{1},h_{2}\in H_{2}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>∈</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}H_{2}=\{h_{1}h_{2}:h_{1}\in H_{1},h_{2}\in H_{2}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a83f9f0fab3051882a75ef101aa989de0b7e97c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.591ex; height:2.843ex;" alt="{\displaystyle H_{1}H_{2}=\{h_{1}h_{2}:h_{1}\in H_{1},h_{2}\in H_{2}\}}"></span>. זוהי תת-חבורה <a href="/wiki/%D7%90%D7%9D_%D7%95%D7%A8%D7%A7_%D7%90%D7%9D" title="אם ורק אם">אם ורק אם</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 H_{1}H_{2}=H_{2}H_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{1}H_{2}=H_{2}H_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a13a0021ca836e8cc6d7a2b1464d197ea5f2fc88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.04ex; height:2.509ex;" alt="{\displaystyle H_{1}H_{2}=H_{2}H_{1}}"></span>. מכיוון שתת-חבורה נורמלית <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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> מקיימת את הזהות <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 xN=Nx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>N</mi> <mo>=</mo> <mi>N</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xN=Nx}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dd0f9f5851ae5cbf82aacebc73873596bebf38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.885ex; height:2.176ex;" alt="{\displaystyle xN=Nx}"></span>, המכפלה שלה עם כל תת-חבורה מהווה תת-חבורה. בפרט, המכפלה של שתי תת-חבורות נורמליות היא תת-חבורה (נורמלית). </p> <div class="mw-heading mw-heading3"><h3 id="תת-חבורות_מיוחדות"><span id=".D7.AA.D7.AA-.D7.97.D7.91.D7.95.D7.A8.D7.95.D7.AA_.D7.9E.D7.99.D7.95.D7.97.D7.93.D7.95.D7.AA"></span>תת-חבורות מיוחדות</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=5" title="עריכת קוד המקור של הפרק: תת-חבורות מיוחדות"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=5" title="עריכת פסקה: "תת-חבורות מיוחדות"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>אומרים שאיברים <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,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"></span> בחבורה <b>מתחלפים</b>, אם <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=ba}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab=ba}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/794f6c310259e816eed4a00262d91bf4f53e37c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.553ex; height:2.176ex;" alt="{\displaystyle ab=ba}"></span>. אוסף האיברים המתחלפים עם כל אברי החבורה הוא תת-חבורה שלה, הנקראת <b><a href="/wiki/%D7%9E%D7%A8%D7%9B%D7%96_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="מרכז (תורת החבורות)">מֶרְכָּז</a></b> החבורה; את המרכז של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> מקובל לסמן ב-<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(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a45cef6993e3ea964232c8014e96998bb0aa3d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.316ex; height:2.843ex;" alt="{\displaystyle Z(G)}"></span>, על-פי המלה הגרמנית למרכז, Zentrum. המרכז הוא תת-חבורה נורמלית, ובחבורה אבלית הוא שווה לחבורה כולה. יש חבורות, כגון <a href="/wiki/%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%94%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="החבורה הסימטרית">החבורה הסימטרית</a>, שבהן המרכז כולל רק את איבר היחידה. המרכז מוכל בכל תת-חבורה אבלית מקסימלית של החבורה. </p><p>באופן כללי יותר, לכל תת-חבורה <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> של חבורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> מסמנים ב-<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 C_{G}(H)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{G}(H)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c14c8b8e3472d2c491df3db342dbf49397b8ad3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.059ex; height:2.843ex;" alt="{\displaystyle C_{G}(H)}"></span> את אוסף האיברים של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, המתחלפים עם כל אברי <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. תת-חבורה זו נקראת ה<b>מְרַכֵּז</b> של <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>. בפרט, <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(G)=C_{G}(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(G)=C_{G}(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4479375a215afcc6987df4d3acabe9d82db1ac5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.237ex; height:2.843ex;" alt="{\displaystyle Z(G)=C_{G}(G)}"></span>. אם <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 H\subseteq H_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊆</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subseteq H_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24a98a2753d50f8979a50d9f2eac9070b43dcdfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.148ex; height:2.509ex;" alt="{\displaystyle H\subseteq H_{1}}"></span> שתי תת-חבורות של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, אז <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 C_{G}(H_{1})\subseteq C_{G}(H)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊆</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{G}(H_{1})\subseteq C_{G}(H)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f6dd9e7298fdeddc6d66fe187e69ff6ca7efdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.138ex; height:2.843ex;" alt="{\displaystyle C_{G}(H_{1})\subseteq C_{G}(H)}"></span>. לכל תת-חבורה מתקיים <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 H\subseteq C_{G}(C_{G}(H))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>⊆</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\subseteq C_{G}(C_{G}(H))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/071b6f7458666563697c5697a5eb7cef562d7c06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.216ex; height:2.843ex;" alt="{\displaystyle H\subseteq C_{G}(C_{G}(H))}"></span>, ו-<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 C_{G}(H)=C_{G}(C_{G}(C_{G}(H)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{G}(H)=C_{G}(C_{G}(C_{G}(H)))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc25381aaa88c1cd17d88291c1eca0004675edd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.206ex; height:2.843ex;" alt="{\displaystyle C_{G}(H)=C_{G}(C_{G}(C_{G}(H)))}"></span>. </p><p>באופן דומה, מגדירים את ה<b>מנרמל</b> של <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, כתת-החבורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{g\in G:gHg^{-1}=H\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∈</mo> <mi>G</mi> <mo>:</mo> <mi>g</mi> <mi>H</mi> <msup> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{g\in G:gHg^{-1}=H\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054099fa428368a8fd3584f98503021e2f398dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.838ex; height:3.176ex;" alt="{\displaystyle \{g\in G:gHg^{-1}=H\}}"></span>. תת-חבורה זו מכילה את <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, והיא תת-החבורה הגדולה ביותר של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> שבה <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 H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> נורמלית. </p><p>ראו גם: <a href="/wiki/%D7%AA%D7%AA-%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%95%D7%9E%D7%95%D7%98%D7%98%D7%95%D7%A8%D7%99%D7%9D" title="תת-חבורת הקומוטטורים">תת-חבורת הקומוטטורים</a>, <a href="/wiki/%D7%A1%D7%93%D7%A8%D7%94_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99%D7%AA" title="סדרה נורמלית">סדרה נורמלית</a>. </p> <div class="mw-heading mw-heading2"><h2 id="יוצרים_ויחסים"><span id=".D7.99.D7.95.D7.A6.D7.A8.D7.99.D7.9D_.D7.95.D7.99.D7.97.D7.A1.D7.99.D7.9D"></span>יוצרים ויחסים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=6" title="עריכת קוד המקור של הפרק: יוצרים ויחסים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=6" title="עריכת פסקה: "יוצרים ויחסים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Postscript-viewer-blue.svg/25px-Postscript-viewer-blue.svg.png" decoding="async" width="25" height="25" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Postscript-viewer-blue.svg/38px-Postscript-viewer-blue.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Postscript-viewer-blue.svg/50px-Postscript-viewer-blue.svg.png 2x" data-file-width="60" data-file-height="60" /></span></span> ערך מורחב – <b><a href="/wiki/%D7%99%D7%99%D7%A6%D7%95%D7%92_%D7%A9%D7%9C_%D7%97%D7%91%D7%95%D7%A8%D7%94" title="ייצוג של חבורה">ייצוג של חבורה</a></b><br /></dd></dl> <p>קבוצה <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> של איברים בחבורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> היא <b>קבוצת יוצרים</b> של <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, אם תת-החבורה הקטנה ביותר המכילה את <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> היא <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> עצמה. חבורה שיש לה קבוצת יוצרים ובה איבר יחיד, נקראת <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A6%D7%99%D7%A7%D7%9C%D7%99%D7%AA" title="חבורה ציקלית">חבורה ציקלית</a>; כל <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A0%D7%95%D7%A6%D7%A8%D7%AA_%D7%A1%D7%95%D7%A4%D7%99%D7%AA" class="mw-redirect" title="חבורה נוצרת סופית">חבורה נוצרת סופית</a> היא <a href="/wiki/%D7%A7%D7%91%D7%95%D7%A6%D7%94_%D7%91%D7%AA_%D7%9E%D7%A0%D7%99%D7%99%D7%94" title="קבוצה בת מנייה">בת-מנייה</a>. </p><p>היוצרים של חבורה יכולים לקיים ביניהם <b>יחסים</b>; למשל, חבורת התמורות של שלושה עצמים נוצרת על ידי התמורות <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 \sigma =(123),\tau =(12)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>123</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>12</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =(123),\tau =(12)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa71334de96de13f5eaf76c28c0479ba8a42314f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.193ex; height:2.843ex;" alt="{\displaystyle \sigma =(123),\tau =(12)}"></span>, המקיימות את היחסים <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 \sigma ^{3}=\tau ^{2}=(\sigma \tau )^{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>σ</mi> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{3}=\tau ^{2}=(\sigma \tau )^{2}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d55e9437d43b4021acd593296fd15c644957dab3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.55ex; height:3.176ex;" alt="{\displaystyle \sigma ^{3}=\tau ^{2}=(\sigma \tau )^{2}=1}"></span>. חבורה שבין היוצרים שלה אין יחסים כלל נקראת <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%97%D7%95%D7%A4%D7%A9%D7%99%D7%AA" title="חבורה חופשית">חבורה חופשית</a>; כל חבורה היא <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%AA_%D7%9E%D7%A0%D7%94" title="חבורת מנה">חבורת מנה</a> של חבורה חופשית. </p><p>את האופן שבו החבורה מכוסה על ידי מילים הולכות ומתארכות מודד קצב ה<b><a href="/wiki/%D7%92%D7%99%D7%93%D7%95%D7%9C_%D7%A9%D7%9C_%D7%97%D7%91%D7%95%D7%A8%D7%94" title="גידול של חבורה">גידול</a></b> של החבורה. לקצב זה יש קשרים הדוקים למבנה של החבורה. </p> <div class="mw-heading mw-heading2"><h2 id="פעולות_יסודיות_ואיברים_צמודים"><span id=".D7.A4.D7.A2.D7.95.D7.9C.D7.95.D7.AA_.D7.99.D7.A1.D7.95.D7.93.D7.99.D7.95.D7.AA_.D7.95.D7.90.D7.99.D7.91.D7.A8.D7.99.D7.9D_.D7.A6.D7.9E.D7.95.D7.93.D7.99.D7.9D"></span>פעולות יסודיות ואיברים צמודים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=7" title="עריכת קוד המקור של הפרק: פעולות יסודיות ואיברים צמודים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=7" title="עריכת פסקה: "פעולות יסודיות ואיברים צמודים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>לחבורות יש מעמד מרכזי במתמטיקה בגלל היכולת שלהן <a href="/wiki/%D7%A4%D7%A2%D7%95%D7%9C%D7%AA_%D7%97%D7%91%D7%95%D7%A8%D7%94" title="פעולת חבורה">לפעול</a> על קבוצות שונות. כמעט בכל מקרה, אוסף הפונקציות ההפיכות ממרחב נתון אל עצמו, השומרות על תכונות מסוימות של המרחב, מהווה חבורה. פעולה של חבורה על קבוצה <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> מציגה את אברי החבורה כפונקציות הפיכות מן הקבוצה <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> לעצמה, וכך מאפשרת לחקור תכונות מעניינות של <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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> מחד, ולנתח ביתר קלות את המבנה של החבורה, מאידך. </p><p>כל חבורה פועלת על עצמה בשתי דרכים חשובות: על ידי פעולת הכפל (משמאל או מימין), ועל ידי פעולת ה<a href="/wiki/%D7%94%D7%A6%D7%9E%D7%93%D7%94_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="הצמדה (תורת החבורות)">הצמדה</a>. פעולת הכפל משמאל מוגדרת באופן שאיבר <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> שולח את האיבר <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> לאיבר <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 xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c72eb345e496513fb8b2fa4aa8c4d89b855f9a01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}"></span>. בדרך זו הופך האיבר <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> ל<a href="/wiki/%D7%AA%D7%9E%D7%95%D7%A8%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="תמורה (מתמטיקה)">תמורה</a> על אברי החבורה, וכך מתקבלת הוכחה של <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98_%D7%A7%D7%99%D7%99%D7%9C%D7%99" title="משפט קיילי">משפט קיילי</a>: כל חבורה היא תת-חבורה של חבורת תמורות. בפעולת ההצמדה, האיבר <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> שולח את <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> ל-<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 xyx^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xyx^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8973312c6f2494a4ef3b7fc6968fef05f0b5136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.148ex; height:3.009ex;" alt="{\displaystyle xyx^{-1}}"></span>; פעולה זו מחלקת את החבורה <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> ל<a href="/wiki/%D7%9E%D7%97%D7%9C%D7%A7%D7%AA_%D7%A9%D7%A7%D7%99%D7%9C%D7%95%D7%AA" title="מחלקת שקילות">מחלקות שקילות</a> מן הצורה <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 \{xyx^{-1}:x\in G\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mi>y</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>x</mi> <mo>∈</mo> <mi>G</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{xyx^{-1}:x\in G\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638b7497efa6e49d2d6666785205119135c8d670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.407ex; height:3.176ex;" alt="{\displaystyle \{xyx^{-1}:x\in G\}}"></span>, הקרויות "מחלקות צמידות". פעולת ההצמדה היא גם <a href="/wiki/%D7%90%D7%95%D7%98%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="אוטומורפיזם">אוטומורפיזם</a> של החבורה עצמה, ולכן היא יוצרת <a href="/wiki/%D7%94%D7%A6%D7%92%D7%94_(%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94)" title="הצגה (מתמטיקה)">הצגה</a> של החבורה, כתת-חבורה של חבורת האוטומורפיזמים של עצמה. הצגה זו היא נאמנה <a href="/wiki/%D7%90%D7%9D_%D7%95%D7%A8%D7%A7_%D7%90%D7%9D" title="אם ורק אם">אם ורק אם</a> לחבורה יש <a href="/wiki/%D7%9E%D7%A8%D7%9B%D7%96_(%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA)" title="מרכז (תורת החבורות)">מרכז</a> טריוויאלי. </p> <div class="mw-heading mw-heading2"><h2 id="ראו_גם"><span id=".D7.A8.D7.90.D7.95_.D7.92.D7.9D"></span>ראו גם</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=8" title="עריכת קוד המקור של הפרק: ראו גם"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=8" title="עריכת פסקה: "ראו גם"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%D7%9E%D7%95%D7%A0%D7%95%D7%90%D7%99%D7%93" class="mw-redirect" title="מונואיד">מונואיד</a></li> <li><a href="/wiki/%D7%97%D7%95%D7%92_%D7%97%D7%91%D7%95%D7%A8%D7%94" title="חוג חבורה">חוג חבורה</a></li> <li><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%AA" title="חבורה טופולוגית">חבורה טופולוגית</a></li> <li><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="חבורה אלגברית">חבורה אלגברית</a></li> <li><a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%A6%D7%99%D7%A7%D7%9C%D7%99%D7%AA" title="חבורה ציקלית">חבורה ציקלית</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="קישורים_חיצוניים"><span id=".D7.A7.D7.99.D7.A9.D7.95.D7.A8.D7.99.D7.9D_.D7.97.D7.99.D7.A6.D7.95.D7.A0.D7.99.D7.99.D7.9D"></span>קישורים חיצוניים</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&action=edit&section=9" title="עריכת קוד המקור של הפרק: קישורים חיצוניים"><span>עריכת קוד מקור</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=%D7%97%D7%91%D7%95%D7%A8%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)&veaction=edit&section=9" title="עריכת פסקה: "קישורים חיצוניים"" class="mw-editsection-visualeditor"><span>עריכה</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="sisterwikilinkT"><span typeof="mw:File"><a href="https://commons.wikimedia.org/wiki/" title="ויקישיתוף"><img alt="ויקישיתוף" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> מדיה וקבצים בנושא <b><a href="https://commons.wikimedia.org/wiki/Category:Group_theory" class="extiw" title="commons:Category:Group theory">חבורה</a></b> ב<a href="/wiki/%D7%95%D7%99%D7%A7%D7%99%D7%A9%D7%99%D7%AA%D7%95%D7%A3" title="ויקישיתוף">וויקישיתוף</a></div> <ul><li><span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/%D7%A1%D7%9E%D7%9C_%D7%95%D7%99%D7%93%D7%99%D7%90%D7%95_%D7%9E%D7%97%D7%95%D7%93%D7%A9_3.svg/16px-%D7%A1%D7%9E%D7%9C_%D7%95%D7%99%D7%93%D7%99%D7%90%D7%95_%D7%9E%D7%97%D7%95%D7%93%D7%A9_3.svg.png" decoding="async" width="16" height="11" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/%D7%A1%D7%9E%D7%9C_%D7%95%D7%99%D7%93%D7%99%D7%90%D7%95_%D7%9E%D7%97%D7%95%D7%93%D7%A9_3.svg/24px-%D7%A1%D7%9E%D7%9C_%D7%95%D7%99%D7%93%D7%99%D7%90%D7%95_%D7%9E%D7%97%D7%95%D7%93%D7%A9_3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/%D7%A1%D7%9E%D7%9C_%D7%95%D7%99%D7%93%D7%99%D7%90%D7%95_%D7%9E%D7%97%D7%95%D7%93%D7%A9_3.svg/32px-%D7%A1%D7%9E%D7%9C_%D7%95%D7%99%D7%93%D7%99%D7%90%D7%95_%D7%9E%D7%97%D7%95%D7%93%D7%A9_3.svg.png 2x" data-file-width="188" data-file-height="131" /></span></span> <a rel="nofollow" class="external text" href="https://youtu.be/mvmuCPvRoWQ">Euler's formula with introductory group theory</a> - סרטון <a href="/wiki/%D7%94%D7%A0%D7%A4%D7%A9%D7%94" title="הנפשה">הנפשה</a></li> <li><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Group.html">חבורה</a>, באתר <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a> <span dir="rtl" class="languageicon">(באנגלית)</span><style data-mw-deduplicate="TemplateStyles:r36549940">.mw-parser-output .languageicon{font-size:0.95em;font-weight:bold;color:#555}</style></li> <li><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/group-mathematics">חבורה</a>, באתר <a href="/wiki/%D7%90%D7%A0%D7%A6%D7%99%D7%A7%D7%9C%D7%95%D7%A4%D7%93%D7%99%D7%94_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="אנציקלופדיה בריטניקה">אנציקלופדיה בריטניקה</a> <span dir="rtl" class="languageicon">(באנגלית)</span><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36549940"></li></ul> <p><br /> </p> <table class="navbox nowraplinks" style="width: 90%; clear: both; margin: 0.5em auto; margin-top: 0.5em; margin-bottom: 0.5em; padding: 0.2em; text-align: right;"> <tbody><tr> <th colspan="3" style="text-align: center; padding-top: 0.1em; padding-bottom: 0.1em; color: black; background:#d1eeee; font-weight: bold;"><a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9E%D7%95%D7%A4%D7%A9%D7%98%D7%AA" title="אלגברה מופשטת">אלגברה מופשטת</a> </th></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">ענפים </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%99%D7%A0%D7%99%D7%90%D7%A8%D7%99%D7%AA" title="אלגברה ליניארית">אלגברה ליניארית</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%91%D7%95%D7%9C%D7%99%D7%90%D7%A0%D7%99%D7%AA" title="אלגברה בוליאנית">אלגברה בוליאנית</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%93%D7%99%D7%A4%D7%A8%D7%A0%D7%A6%D7%99%D7%90%D7%9C%D7%99%D7%AA" title="אלגברה דיפרנציאלית">אלגברה דיפרנציאלית</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%94%D7%95%D7%9E%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%AA" title="אלגברה הומולוגית">אלגברה הומולוגית</a> • <a href="/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="גאומטריה אלגברית">גאומטריה אלגברית</a> • <a href="/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="טופולוגיה אלגברית">טופולוגיה אלגברית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%92%D7%9C%D7%95%D7%90%D7%94" title="תורת גלואה">תורת גלואה</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA" title="תורת החבורות">תורת החבורות</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%95%D7%92%D7%99%D7%9D" title="תורת החוגים">תורת החוגים</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%9E%D7%A1%D7%A4%D7%A8%D7%99%D7%9D_%D7%94%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="תורת המספרים האלגברית">תורת המספרים האלגברית</a> • <a href="/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%95%D7%AA" title="תורת הקטגוריות">תורת הקטגוריות</a> • <a href="/w/index.php?title=%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%A9%D7%93%D7%95%D7%AA&action=edit&redlink=1" class="new" title="תורת השדות (הדף אינו קיים)">תורת השדות</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;"><a href="/wiki/%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99" title="מבנה אלגברי">מבנים אלגבריים</a> </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%9E%D7%90%D7%92%D7%9E%D7%94_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מאגמה (מבנה אלגברי)">מאגמה</a> • <a href="/wiki/%D7%97%D7%91%D7%95%D7%A8%D7%94_%D7%9C%D7%9E%D7%97%D7%A6%D7%94" title="חבורה למחצה">חבורה למחצה</a> • <a href="/wiki/%D7%9E%D7%95%D7%A0%D7%95%D7%90%D7%99%D7%93_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="מונואיד (מבנה אלגברי)">מונואיד</a> • <a class="mw-selflink selflink">חבורה</a> • <a 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title="אלגברת לי">אלגברת לי</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%AA_%D7%94%D7%A7%D7%95%D7%95%D7%98%D7%A8%D7%A0%D7%99%D7%95%D7%A0%D7%99%D7%9D_%D7%A9%D7%9C_%D7%94%D7%9E%D7%99%D7%9C%D7%98%D7%95%D7%9F" title="אלגברת הקווטרניונים של המילטון">אלגברת הקווטרניונים של המילטון</a> • <a href="/wiki/%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94_%D7%9C%D7%90_%D7%90%D7%A1%D7%95%D7%A6%D7%99%D7%90%D7%98%D7%99%D7%91%D7%99%D7%AA" title="אלגברה לא אסוציאטיבית">אלגברה לא אסוציאטיבית</a> </td></tr> <tr> <td style="background-color: #F2F3F4; text-align: right; font-weight: bold; padding-left: 5px;">מושגי יסוד </td> <td style="padding-right: 5px; text-align: right;"><a href="/wiki/%D7%94%D7%95%D7%9E%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="הומומורפיזם">הומומורפיזם</a> • <a href="/wiki/%D7%9E%D7%A9%D7%A4%D7%98%D7%99_%D7%94%D7%90%D7%99%D7%96%D7%95%D7%9E%D7%95%D7%A8%D7%A4%D7%99%D7%96%D7%9D" title="משפטי האיזומורפיזם">משפטי האיזומורפיזם</a> • <a 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style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th id="בקרת_זהויות_15px&#124;link=https&#58;//www.wikidata.org/wiki/Q83478?uselang=he&#124;עריכת_הנתון_בוויקינתונים" scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%D7%A2%D7%96%D7%A8%D7%94:%D7%91%D7%A7%D7%A8%D7%AA_%D7%96%D7%94%D7%95%D7%99%D7%95%D7%AA" title="עזרה:בקרת זהויות">בקרת זהויות</a> <span typeof="mw:File"><a href="https://www.wikidata.org/wiki/Q83478?uselang=he" title="עריכת הנתון בוויקינתונים"><img alt="עריכת הנתון בוויקינתונים" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/15px-Blue_pencil_RTL.svg.png" decoding="async" width="15" height="15" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/23px-Blue_pencil_RTL.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Blue_pencil_RTL.svg/30px-Blue_pencil_RTL.svg.png 2x" data-file-width="600" data-file-height="600" /></a></span></th><td class="navbox-list navbox-odd" style="text-align:left;border-right-width:2px;border-right-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"><div dir="ltr"> <ul><li><span class="nowrap"><a href="/wiki/%D7%A7%D7%95%D7%91%D7%A5_%D7%91%D7%A7%D7%A8%D7%94_%D7%9E%D7%A9%D7%95%D7%9C%D7%91" title="קובץ בקרה משולב">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4022379-6">4022379-6</a></span></span></li> <li><span class="nowrap"><a href="/wiki/%D7%94%D7%A1%D7%A4%D7%A8%D7%99%D7%99%D7%94_%D7%94%D7%9C%D7%90%D7%95%D7%9E%D7%99%D7%AA_%D7%A9%D7%9C_%D7%A6%27%D7%9B%D7%99%D7%94" title="הספרייה הלאומית של צ'כיה">NKC</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph180740&CON_LNG=ENG">ph180740</a></span></span></li></ul> </div></div></td></tr></tbody></table></div>    </div><noscript><img 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href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA" title="קטגוריה:תורת החבורות">תורת החבורות</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%91%D7%9E%D7%AA%D7%9E%D7%98%D7%99%D7%A7%D7%94" title="קטגוריה:סימטריה במתמטיקה">סימטריה במתמטיקה</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">קטגוריות מוסתרות: <ul><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%AA%D7%91%D7%A0%D7%99%D7%AA_%D7%91%D7%A8%D7%99%D7%98%D7%A0%D7%99%D7%A7%D7%94" title="קטגוריה:ערכים עם תבנית בריטניקה">ערכים עם תבנית בריטניקה</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_GND" title="קטגוריה:ויקיפדיה: ערכים עם מזהה GND">ויקיפדיה: ערכים עם מזהה GND</a></li><li><a href="/wiki/%D7%A7%D7%98%D7%92%D7%95%D7%A8%D7%99%D7%94:%D7%95%D7%99%D7%A7%D7%99%D7%A4%D7%93%D7%99%D7%94:_%D7%A2%D7%A8%D7%9B%D7%99%D7%9D_%D7%A2%D7%9D_%D7%9E%D7%96%D7%94%D7%94_NKC" title="קטגוריה:ויקיפדיה: ערכים עם מזהה NKC">ויקיפדיה: ערכים עם מזהה NKC</a></li></ul></div></div> </div> </main> <div id='mw-data-after-content'> <div class="read-more-container"></div> </div> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> דף זה נערך לאחרונה ב־8 בספטמבר 2024, בשעה 19:52.</li> <li id="footer-info-copyright">הטקסט מוגש בכפוף לרישיון <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.he">Creative Commons ייחוס-שיתוף זהה 4.0</a>; ייתכן שישנם תנאים נוספים. ר׳ את <a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">תנאי השימוש</a> לפרטים.</li> </ul> <ul id="footer-places"> <li 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