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vector-toc-level-2"> <a class="vector-toc-link" href="#Koncepte_ne_teorinë_e_grupeve"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Koncepte ne teorinë e grupeve</span> </div> </a> <ul id="toc-Koncepte_ne_teorinë_e_grupeve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rendi_i_grupit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rendi_i_grupit"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Rendi i grupit</span> </div> </a> <ul id="toc-Rendi_i_grupit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Shih_edhe" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Shih_edhe"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Shih edhe</span> </div> </a> <ul id="toc-Shih_edhe-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lidhje_të_jashtme" 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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">Toggle the table of contents</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">Teoria e grupeve</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="Go to an article in another language. Available in 72 languages" > <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-72" 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">72 gjuhë</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – aragonezisht" lang="an" hreflang="an" data-title="Teoría de grupos" data-language-autonym="Aragonés" data-language-local-name="aragonezisht" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%B2%D9%85%D8%B1" title="نظرية الزمر – arabisht" lang="ar" hreflang="ar" data-title="نظرية الزمر" data-language-autonym="العربية" data-language-local-name="arabisht" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – asturisht" lang="ast" hreflang="ast" data-title="Teoría de grupos" data-language-autonym="Asturianu" data-language-local-name="asturisht" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Qrup_n%C9%99z%C9%99riyy%C9%99si" title="Qrup nəzəriyyəsi – azerbajxhanisht" lang="az" hreflang="az" data-title="Qrup nəzəriyyəsi" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajxhanisht" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%A2%D3%A9%D1%80%D0%BA%D3%A9%D0%BC%D0%B4%D3%99%D1%80_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F%D2%BB%D1%8B" title="Төркөмдәр теорияһы – bashkirisht" lang="ba" hreflang="ba" data-title="Төркөмдәр теорияһы" data-language-autonym="Башҡортса" data-language-local-name="bashkirisht" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bat-smg mw-list-item"><a href="https://bat-smg.wikipedia.org/wiki/Gropiu_teuor%C4%97j%C4%97" title="Gropiu teuorėjė – Samogitian" lang="sgs" hreflang="sgs" data-title="Gropiu teuorėjė" data-language-autonym="Žemaitėška" data-language-local-name="Samogitian" class="interlanguage-link-target"><span>Žemaitėška</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B3%D1%80%D1%83%D0%BF" title="Тэорыя груп – bjellorusisht" lang="be" hreflang="be" data-title="Тэорыя груп" data-language-autonym="Беларуская" data-language-local-name="bjellorusisht" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A2%D1%8D%D0%BE%D1%80%D1%8B%D1%8F_%D0%B3%D1%80%D1%83%D0%BF%D0%B0%D1%9E" title="Тэорыя групаў – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Тэорыя групаў" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B8%D1%82%D0%B5" title="Теория на групите – bullgarisht" lang="bg" hreflang="bg" data-title="Теория на групите" data-language-autonym="Български" data-language-local-name="bullgarisht" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Damkaniezh_ar_strollo%C3%B9" title="Damkaniezh ar strolloù – bretonisht" lang="br" hreflang="br" data-title="Damkaniezh ar strolloù" data-language-autonym="Brezhoneg" data-language-local-name="bretonisht" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Teorija_grupa" title="Teorija grupa – boshnjakisht" lang="bs" hreflang="bs" data-title="Teorija grupa" data-language-autonym="Bosanski" data-language-local-name="boshnjakisht" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Teoria_de_grups" title="Teoria de grups – katalonisht" lang="ca" hreflang="ca" data-title="Teoria de grups" data-language-autonym="Català" data-language-local-name="katalonisht" 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/%D8%AA%DB%8C%DB%86%D8%B1%DB%8C%DB%8C_%DA%AF%D8%B1%D9%88%D9%88%D9%BE" title="تیۆریی گرووپ – kurdishte qendrore" lang="ckb" hreflang="ckb" data-title="تیۆریی گرووپ" data-language-autonym="کوردی" data-language-local-name="kurdishte qendrore" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Teorie_grup" title="Teorie grup – çekisht" lang="cs" hreflang="cs" data-title="Teorie grup" data-language-autonym="Čeština" data-language-local-name="çekisht" 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%D1%81%D0%B5%D0%BD_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D0%B9%C4%95" title="Ушкăнсен теорийĕ – çuvashisht" lang="cv" hreflang="cv" data-title="Ушкăнсен теорийĕ" data-language-autonym="Чӑвашла" data-language-local-name="çuvashisht" 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/Damcaniaeth_grwpiau" title="Damcaniaeth grwpiau – uellsisht" lang="cy" hreflang="cy" data-title="Damcaniaeth grwpiau" data-language-autonym="Cymraeg" data-language-local-name="uellsisht" 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/Gruppeteori_(matematik)" title="Gruppeteori (matematik) – danisht" lang="da" hreflang="da" data-title="Gruppeteori (matematik)" data-language-autonym="Dansk" data-language-local-name="danisht" 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/Gruppentheorie" title="Gruppentheorie – gjermanisht" lang="de" hreflang="de" data-title="Gruppentheorie" data-language-autonym="Deutsch" data-language-local-name="gjermanisht" 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%98%CE%B5%CF%89%CF%81%CE%AF%CE%B1_%CE%BF%CE%BC%CE%AC%CE%B4%CF%89%CE%BD" title="Θεωρία ομάδων – greqisht" lang="el" hreflang="el" data-title="Θεωρία ομάδων" data-language-autonym="Ελληνικά" data-language-local-name="greqisht" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Group_theory" title="Group theory – anglisht" lang="en" hreflang="en" data-title="Group theory" data-language-autonym="English" data-language-local-name="anglisht" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Grupo-teorio" title="Grupo-teorio – esperanto" lang="eo" hreflang="eo" data-title="Grupo-teorio" data-language-autonym="Esperanto" data-language-local-name="esperanto" 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/Teor%C3%ADa_de_grupos" title="Teoría de grupos – spanjisht" lang="es" hreflang="es" data-title="Teoría de grupos" data-language-autonym="Español" data-language-local-name="spanjisht" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Talde-teoria" title="Talde-teoria – baskisht" lang="eu" hreflang="eu" data-title="Talde-teoria" data-language-autonym="Euskara" data-language-local-name="baskisht" 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/%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%DA%AF%D8%B1%D9%88%D9%87%E2%80%8C%D9%87%D8%A7" title="نظریه گروه‌ها – persisht" lang="fa" hreflang="fa" data-title="نظریه گروه‌ها" data-language-autonym="فارسی" data-language-local-name="persisht" 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" title="Ryhmä – finlandisht" lang="fi" hreflang="fi" data-title="Ryhmä" data-language-autonym="Suomi" data-language-local-name="finlandisht" 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/Th%C3%A9orie_des_groupes" title="Théorie des groupes – frëngjisht" lang="fr" hreflang="fr" data-title="Théorie des groupes" data-language-autonym="Français" data-language-local-name="frëngjisht" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fur mw-list-item"><a href="https://fur.wikipedia.org/wiki/Teorie_dai_grups" title="Teorie dai grups – friulianisht" lang="fur" hreflang="fur" data-title="Teorie dai grups" data-language-autonym="Furlan" data-language-local-name="friulianisht" class="interlanguage-link-target"><span>Furlan</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Teor%C3%ADa_de_grupos" title="Teoría de grupos – galicisht" lang="gl" hreflang="gl" data-title="Teoría de grupos" data-language-autonym="Galego" data-language-local-name="galicisht" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%95%D7%A8%D7%AA_%D7%94%D7%97%D7%91%D7%95%D7%A8%D7%95%D7%AA" title="תורת החבורות – hebraisht" lang="he" hreflang="he" data-title="תורת החבורות" data-language-autonym="עברית" data-language-local-name="hebraisht" class="interlanguage-link-target"><span>עברית</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%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%82%E0%A4%A4" title="समूह सिद्धांत – indisht" lang="hi" hreflang="hi" data-title="समूह सिद्धांत" data-language-autonym="हिन्दी" data-language-local-name="indisht" 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/Teorija_grupa" title="Teorija grupa – kroatisht" lang="hr" hreflang="hr" data-title="Teorija grupa" data-language-autonym="Hrvatski" data-language-local-name="kroatisht" 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/Csoportelm%C3%A9let" title="Csoportelmélet – hungarisht" lang="hu" hreflang="hu" data-title="Csoportelmélet" data-language-autonym="Magyar" data-language-local-name="hungarisht" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Theoria_de_gruppos" title="Theoria de gruppos – interlingua" lang="ia" hreflang="ia" data-title="Theoria de gruppos" data-language-autonym="Interlingua" data-language-local-name="interlingua" 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/Teori_grup" title="Teori grup – indonezisht" lang="id" hreflang="id" data-title="Teori grup" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezisht" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Teoria_dei_gruppi" title="Teoria dei gruppi – italisht" lang="it" hreflang="it" data-title="Teoria dei gruppi" data-language-autonym="Italiano" data-language-local-name="italisht" 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%E8%AB%96" title="群論 – japonisht" lang="ja" hreflang="ja" data-title="群論" data-language-autonym="日本語" data-language-local-name="japonisht" 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%97%E1%83%90_%E1%83%97%E1%83%94%E1%83%9D%E1%83%A0%E1%83%98%E1%83%90" title="ჯგუფთა თეორია – gjeorgjisht" lang="ka" hreflang="ka" data-title="ჯგუფთა თეორია" data-language-autonym="ქართული" data-language-local-name="gjeorgjisht" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B0%EB%A1%A0" title="군론 – koreanisht" lang="ko" hreflang="ko" data-title="군론" data-language-autonym="한국어" data-language-local-name="koreanisht" 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/Theoria_catervarum" title="Theoria catervarum – latinisht" lang="la" hreflang="la" data-title="Theoria catervarum" data-language-autonym="Latina" data-language-local-name="latinisht" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Grupu_teorija" title="Grupu teorija – letonisht" lang="lv" hreflang="lv" data-title="Grupu teorija" data-language-autonym="Latviešu" data-language-local-name="letonisht" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Teori_kumpulan" title="Teori kumpulan – malajisht" lang="ms" hreflang="ms" data-title="Teori kumpulan" data-language-autonym="Bahasa Melayu" data-language-local-name="malajisht" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%97%E0%A5%8D%E0%A4%B0%E0%A5%81%E0%A4%AA_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="ग्रुप सिद्धान्त – neuarisht" lang="new" hreflang="new" data-title="ग्रुप सिद्धान्त" data-language-autonym="नेपाल भाषा" data-language-local-name="neuarisht" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Groepentheorie" title="Groepentheorie – holandisht" lang="nl" hreflang="nl" data-title="Groepentheorie" data-language-autonym="Nederlands" data-language-local-name="holandisht" 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/Gruppeteori" title="Gruppeteori – norvegjishte nynorsk" lang="nn" hreflang="nn" data-title="Gruppeteori" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegjishte nynorsk" 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) – norvegjishte letrare" lang="nb" hreflang="nb" data-title="Gruppe (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norvegjishte letrare" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A8%B0%E0%A9%81%E0%A9%B1%E0%A8%AA_%E0%A8%A5%E0%A8%BF%E0%A8%8A%E0%A8%B0%E0%A9%80" title="ਗਰੁੱਪ ਥਿਊਰੀ – punxhabisht" lang="pa" hreflang="pa" data-title="ਗਰੁੱਪ ਥਿਊਰੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punxhabisht" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Teoria_grup" title="Teoria grup – polonisht" lang="pl" hreflang="pl" data-title="Teoria grup" data-language-autonym="Polski" data-language-local-name="polonisht" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%B9%D9%88%D9%84%DB%8C_%D8%B3%D9%88%DA%86" title="ٹولی سوچ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ٹولی سوچ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" 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/Teoria_dos_grupos" title="Teoria dos grupos – portugalisht" lang="pt" hreflang="pt" data-title="Teoria dos grupos" data-language-autonym="Português" data-language-local-name="portugalisht" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Teoria_grupurilor" title="Teoria grupurilor – rumanisht" lang="ro" hreflang="ro" data-title="Teoria grupurilor" data-language-autonym="Română" data-language-local-name="rumanisht" 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" title="Группа – rusisht" lang="ru" hreflang="ru" data-title="Группа" data-language-autonym="Русский" data-language-local-name="rusisht" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D2%91%D1%80%D1%83%D0%BF" title="Теорія ґруп – Rusyn" lang="rue" hreflang="rue" data-title="Теорія ґруп" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target"><span>Русиньскый</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Group_theory" title="Group theory – skotisht" lang="sco" hreflang="sco" data-title="Group theory" data-language-autonym="Scots" data-language-local-name="skotisht" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Teorija_grupa" title="Teorija grupa – serbo-kroatisht" lang="sh" hreflang="sh" data-title="Teorija grupa" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbo-kroatisht" 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_theory" title="Group theory – Simple English" lang="en-simple" hreflang="en-simple" data-title="Group theory" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Grupa" title="Grupa – sllovakisht" lang="sk" hreflang="sk" data-title="Grupa" data-language-autonym="Slovenčina" data-language-local-name="sllovakisht" 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_(matematika)" title="grupa (matematika) – sllovenisht" lang="sl" hreflang="sl" data-title="grupa (matematika)" data-language-autonym="Slovenščina" data-language-local-name="sllovenisht" 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%A2%D0%B5%D0%BE%D1%80%D0%B8%D1%98%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Теорија група – serbisht" lang="sr" hreflang="sr" data-title="Теорија група" data-language-autonym="Српски / srpski" data-language-local-name="serbisht" 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/Gruppteori" title="Gruppteori – suedisht" lang="sv" hreflang="sv" data-title="Gruppteori" data-language-autonym="Svenska" data-language-local-name="suedisht" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AF%81%E0%AE%B2%E0%AE%95%E0%AF%8D_%E0%AE%95%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%AA%E0%AE%BE%E0%AE%9F%E0%AF%81" title="குலக் கோட்பாடு – tamilisht" lang="ta" hreflang="ta" data-title="குலக் கோட்பாடு" data-language-autonym="தமிழ்" data-language-local-name="tamilisht" class="interlanguage-link-target"><span>தமிழ்</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%B9%8A%E0%B8%9B" title="กรุ๊ป – tajlandisht" lang="th" hreflang="th" data-title="กรุ๊ป" data-language-autonym="ไทย" data-language-local-name="tajlandisht" 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/Teorya_ng_grupo" title="Teorya ng grupo – Tagalog" lang="tl" hreflang="tl" data-title="Teorya ng grupo" data-language-autonym="Tagalog" data-language-local-name="Tagalog" 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_teorisi" title="Grup teorisi – turqisht" lang="tr" hreflang="tr" data-title="Grup teorisi" data-language-autonym="Türkçe" data-language-local-name="turqisht" 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%A2%D0%B5%D0%BE%D1%80%D1%96%D1%8F_%D0%B3%D1%80%D1%83%D0%BF" title="Теорія груп – ukrainisht" lang="uk" hreflang="uk" data-title="Теорія груп" data-language-autonym="Українська" data-language-local-name="ukrainisht" 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/%D9%86%D8%B8%D8%B1%DB%8C%DB%82_%DA%AF%D8%B1%D9%88%DB%81" title="نظریۂ گروہ – urduisht" lang="ur" hreflang="ur" data-title="نظریۂ گروہ" data-language-autonym="اردو" data-language-local-name="urduisht" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%C3%BD_thuy%E1%BA%BFt_nh%C3%B3m" title="Lý thuyết nhóm – vietnamisht" lang="vi" hreflang="vi" data-title="Lý thuyết nhóm" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamisht" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Teyorya_grupo" title="Teyorya grupo – uarajisht" lang="war" hreflang="war" data-title="Teyorya grupo" data-language-autonym="Winaray" data-language-local-name="uarajisht" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%BE%A4%E8%AE%BA" title="群论 – kinezishte vu" lang="wuu" hreflang="wuu" data-title="群论" data-language-autonym="吴语" data-language-local-name="kinezishte vu" 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%98%D7%A2%D7%90%D7%A8%D7%99%D7%A2" title="גרופע טעאריע – jidisht" lang="yi" hreflang="yi" data-title="גרופע טעאריע" data-language-autonym="ייִדיש" data-language-local-name="jidisht" 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 – kinezisht" lang="zh" hreflang="zh" data-title="群" data-language-autonym="中文" data-language-local-name="kinezisht" class="interlanguage-link-target"><span>中文</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-l%C5%ABn" title="Kûn-lūn – Minnan" lang="nan" hreflang="nan" data-title="Kûn-lūn" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" 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%E8%AB%96" title="羣論 – kantonezisht" lang="yue" hreflang="yue" data-title="羣論" data-language-autonym="粵語" data-language-local-name="kantonezisht" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit 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Punime të çmueshme në teorinë e grupeve kanë dhënë ndër të tjerë <a href="/w/index.php?title=Evariste_Galois&action=edit&redlink=1" class="new" title="Evariste Galois (nuk është shkruar akoma)">Evariste Galois</a>, <a href="/w/index.php?title=Niels_Henrik_Abel&action=edit&redlink=1" class="new" title="Niels Henrik Abel (nuk është shkruar akoma)">Niels Henrik Abel</a>, <a href="/w/index.php?title=Sophus_Lie&action=edit&redlink=1" class="new" title="Sophus Lie (nuk është shkruar akoma)">Sophus Lie</a> etj. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Përkufizimi_i_grupit"><span id="P.C3.ABrkufizimi_i_grupit"></span>Përkufizimi i grupit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_e_grupeve&veaction=edit&section=1" title="Redakto pjesën: Përkufizimi i grupit" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_e_grupeve&action=edit&section=1" title="Edit section's source code: Përkufizimi i grupit"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Çifti i renditur (G, ×), ku G është një bashkësi dhe × është një <a href="/wiki/Veprim_i_brendsh%C3%ABm" title="Veprim i brendshëm">veprim i brendshëm</a> mbi G, quhet <b>grup</b>, në qoftë se plotësohen këto <a href="/wiki/Aksioma" title="Aksioma">aksioma</a> : </p> <ol><li><a href="/w/index.php?title=Vetia_asociative&action=edit&redlink=1" class="new" title="Vetia asociative (nuk është shkruar akoma)">vetia asociative</a>: a × (b × c) = (a × b) × c për të gjithë elementet a, b, c nga G.</li> <li><a href="/w/index.php?title=Elementi_neutral&action=edit&redlink=1" class="new" title="Elementi neutral (nuk është shkruar akoma)">elementi neutral</a> / (<a href="/w/index.php?title=Elementi_nj%C3%ABsi&action=edit&redlink=1" class="new" title="Elementi njësi (nuk është shkruar akoma)">elementi njësi</a>): Një element <i>e</i> (quhet edhe njësh, element njësi) në G ekziston, i tillë që për të gjithë elementet a nga G vlen vetia <i>e</i> × a = a .</li> <li><a href="/w/index.php?title=Elementi_inverz&action=edit&redlink=1" class="new" title="Elementi inverz (nuk është shkruar akoma)">elementi inverz</a> / (<a href="/w/index.php?title=I_anasjellt%C3%AB&action=edit&redlink=1" class="new" title="I anasjelltë (nuk është shkruar akoma)">i anasjelltë</a>): Për çdo element a në G ekziston një element b, i tillë që vlen barazimi a × b = b × a = <i>e</i> .</li></ol> <p>(Nganjëherë merret si aksiomë e parë "aksioma e mbylljes" sipas së cilës për çdo dy elemente a,b nga G, atëherë a × b është poashtu në G. Mirëpo, kjo "aksiomë" rrjedh nga përkufizimi i veprimit të brendshëm, andaj nuk ka nevojë të shkruhet si e veçantë.) </p><p>Në qoftë se në grupin (G, ×) vlen a × b = b × a, për çdo element a, b nga G, atëherë (G, ×) quhet grup abelian (ose ose komutativ). </p><p>Direkt nga aksiomat e grupit rrjedhin këto pohime: </p><p>Në qoftë se për elementin neutral e të grupit G plotësohet ekuacioni <i>e</i> x a = a për çdo a nga G, atëherë plotësohet edhe ekuacioni a x <i>e</i> = a : a = <i>e</i> × a = (a × b) × a = a × ( b × a) = a × <i>e</i> , kështu që a = <i>e</i> × a = a × <i>e</i>. </p><p>Në çdo grup elementi neutral është i vetëm: Le të jenë e edhe e' dy elemente neutrale në grupin G. Atëherë ndjek qe e = e × e' = e', kështu qe e = e'. </p><p>Pra ne mund të flasim për elementin neutral të grupit G. </p><p>Për çdo element a, elementi simetrik për të është i vetëm, sepse: Le të jenë b dhe b' dy elemente simetrike për elementin a, kështu që a × b = b × a = <i>e</i> dhe a × b' = b' × a = <i>e</i>. Atëherë ndjek që b = b × <i>e</i> = b × ( a × b') = (b × a) × b' = <i>e</i> × b' = b'. Prandaj ne mund të flasim për elementin simetrik të elementit a. </p><p>Në qoftë se veprimi × mbi grupin G është shënuar me mënyrën e shumëzimit, atëherë zakonisht elementi neutral shënohet me 1, kurse elementi simetrik për a shënohet me a<sup>-1</sup>. Kurse, në rastet kur veprimi × mbi grupin G është shënuar me mënyrën e mbledhjes, atëherë zakonisht elementi neutral shënohet me 0, kurse elementi inverz për a shënohet me -a. </p><p>Duhet theksuar se zakonisht × shënohet me mënyrën e mbledhjes kur kemi të bëjmë me një grup abelian. </p> <div class="mw-heading mw-heading3"><h3 id="Koncepte_ne_teorinë_e_grupeve"><span id="Koncepte_ne_teorin.C3.AB_e_grupeve"></span>Koncepte ne teorinë e grupeve</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_e_grupeve&veaction=edit&section=2" title="Redakto pjesën: Koncepte ne teorinë e grupeve" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_e_grupeve&action=edit&section=2" title="Edit section's source code: Koncepte ne teorinë e grupeve"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Rendi_i_grupit">Rendi i grupit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_e_grupeve&veaction=edit&section=3" title="Redakto pjesën: Rendi i grupit" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_e_grupeve&action=edit&section=3" title="Edit section's source code: Rendi i grupit"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Rendi |G| i një grupi (G,×) është numri i elementeve të bashkësisë G. </p> <div class="mw-heading mw-heading2"><h2 id="Shih_edhe">Shih edhe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_e_grupeve&veaction=edit&section=4" title="Redakto pjesën: Shih edhe" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_e_grupeve&action=edit&section=4" title="Edit section's source code: Shih edhe"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bashk%C3%ABsit%C3%AB" title="Bashkësitë">Bashkësitë</a></li> <li><a href="/wiki/Ndihm%C3%AB:Formula" class="mw-redirect" title="Ndihmë:Formula">Ndihmë:Formula</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lidhje_të_jashtme"><span id="Lidhje_t.C3.AB_jashtme"></span>Lidhje të jashtme</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Teoria_e_grupeve&veaction=edit&section=5" title="Redakto pjesën: Lidhje të jashtme" class="mw-editsection-visualeditor"><span>Redakto</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Teoria_e_grupeve&action=edit&section=5" title="Edit section's source code: Lidhje të jashtme"><span>Redakto nëpërmjet kodit</span></a><span class="mw-editsection-bracket">]</span></span></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Marrë nga "<a dir="ltr" href="https://sq.wikipedia.org/w/index.php?title=Teoria_e_grupeve&oldid=2703786">https://sq.wikipedia.org/w/index.php?title=Teoria_e_grupeve&oldid=2703786</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Speciale:Kategorit%C3%AB" title="Speciale:Kategoritë">Kategoria</a>: <ul><li><a href="/wiki/Kategoria:Teorema" title="Kategoria:Teorema">Teorema</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Kjo faqe është redaktuar për herë te fundit më 9 tetor 2024, në orën 10:16.</li> <li id="footer-info-copyright">Të gjitha materialet që gjenden në këtë faqë janë të mbrojtura nga <a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/">Creative Commons Attribution/Share-Alike License</a>;. 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