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Disponible en 14 langues." > <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-14" 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">14 langues</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Grup_lineal_especial" title="Grup lineal especial – catalan" lang="ca" hreflang="ca" data-title="Grup lineal especial" data-language-autonym="Català" data-language-local-name="catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Speci%C3%A1ln%C3%AD_line%C3%A1rn%C3%AD_grupa" title="Speciální lineární grupa – tchèque" lang="cs" hreflang="cs" data-title="Speciální lineární grupa" data-language-autonym="Čeština" data-language-local-name="tchèque" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spezielle_lineare_Gruppe" title="Spezielle lineare Gruppe – allemand" lang="de" hreflang="de" data-title="Spezielle lineare Gruppe" data-language-autonym="Deutsch" data-language-local-name="allemand" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Special_linear_group" title="Special linear group – anglais" lang="en" hreflang="en" data-title="Special linear group" data-language-autonym="English" data-language-local-name="anglais" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Grupo_lineal_especial" title="Grupo lineal especial – espagnol" lang="es" hreflang="es" data-title="Grupo lineal especial" data-language-autonym="Español" data-language-local-name="espagnol" class="interlanguage-link-target"><span>Español</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%AE%D8%B7%DB%8C_%D8%AE%D8%A7%D8%B5" title="گروه خطی خاص – persan" lang="fa" hreflang="fa" data-title="گروه خطی خاص" data-language-autonym="فارسی" data-language-local-name="persan" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Speci%C3%A1lis_line%C3%A1ris_csoport" title="Speciális lineáris csoport – hongrois" lang="hu" hreflang="hu" data-title="Speciális lineáris csoport" data-language-autonym="Magyar" data-language-local-name="hongrois" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%89%B9%E6%AE%8A%E7%B7%9A%E5%9E%8B%E7%BE%A4" title="特殊線型群 – japonais" lang="ja" hreflang="ja" data-title="特殊線型群" data-language-autonym="日本語" data-language-local-name="japonais" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8A%B9%EC%88%98%EC%84%A0%ED%98%95%EA%B5%B0" title="특수선형군 – coréen" lang="ko" hreflang="ko" data-title="특수선형군" data-language-autonym="한국어" data-language-local-name="coréen" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Speciale_lineaire_groep" title="Speciale lineaire groep – néerlandais" lang="nl" hreflang="nl" data-title="Speciale lineaire groep" data-language-autonym="Nederlands" data-language-local-name="néerlandais" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pe%C5%82na_grupa_liniowa#Specjalna_grupa_liniowa" title="Pełna grupa liniowa – polonais" lang="pl" hreflang="pl" data-title="Pełna grupa liniowa" data-language-autonym="Polski" data-language-local-name="polonais" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Grupo_linear_especial" title="Grupo linear especial – portugais" lang="pt" hreflang="pt" data-title="Grupo linear especial" data-language-autonym="Português" data-language-local-name="portugais" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-uk badge-Q70893996 mw-list-item" title=""><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%BB%D1%96%D0%BD%D1%96%D0%B9%D0%BD%D0%B0_%D0%B3%D1%80%D1%83%D0%BF%D0%B0" title="Спеціальна лінійна група – ukrainien" lang="uk" hreflang="uk" data-title="Спеціальна лінійна група" data-language-autonym="Українська" data-language-local-name="ukrainien" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%B8%80%E8%88%AC%E7%BA%BF%E6%80%A7%E7%BE%A4#特殊線性群" title="一般线性群 – chinois" lang="zh" hreflang="zh" data-title="一般线性群" data-language-autonym="中文" data-language-local-name="chinois" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2140900#sitelinks-wikipedia" title="Modifier les liens interlangues" class="wbc-editpage">Modifier les liens</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Espaces de noms"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Groupe_sp%C3%A9cial_lin%C3%A9aire" title="Voir le contenu de la page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Discussion:Groupe_sp%C3%A9cial_lin%C3%A9aire" rel="discussion" title="Discussion au sujet de cette page de contenu [t]" accesskey="t"><span>Discussion</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><div class="bandeau-container metadata bandeau-article bandeau-niveau-ebauche"><div class="bandeau-cell bandeau-icone" style="display:table-cell;padding-right:0.5em"><span class="noviewer" typeof="mw:File"><a href="/wiki/Fichier:Evariste_galois.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Evariste_galois.jpg/27px-Evariste_galois.jpg" decoding="async" width="27" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Evariste_galois.jpg/41px-Evariste_galois.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Evariste_galois.jpg/54px-Evariste_galois.jpg 2x" data-file-width="792" data-file-height="1024" /></a></span></div><div class="bandeau-cell" style="display:table-cell;padding-right:0.5em"> <p><strong class="bandeau-titre">Cet article est une <a href="/wiki/Aide:%C3%89bauche" title="Aide:Ébauche">ébauche</a> concernant l’<a href="/wiki/Alg%C3%A8bre" title="Algèbre">algèbre</a>.</strong> </p><p>Vous pouvez partager vos connaissances en l’améliorant (<b><a href="/wiki/Aide:Comment_modifier_une_page" title="Aide:Comment modifier une page">comment ?</a></b>) selon les recommandations des <a href="/wiki/Projet:Accueil" title="Projet:Accueil">projets correspondants</a>. </p> </div></div> <p>En <a href="/wiki/Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, le <b>groupe spécial linéaire</b> de degré <i>n</i> sur un <a href="/wiki/Corps_commutatif" title="Corps commutatif">corps commutatif</a> <i>K</i> est le <a href="/wiki/Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a> <i>SL</i><sub><i>n</i></sub>(<i>K</i>) des matrices carrées d'ordre n sur <i>K</i> dont le <a href="/wiki/D%C3%A9terminant_(math%C3%A9matiques)" title="Déterminant (mathématiques)">déterminant</a> est égal à 1. Plus intrinsèquement, le groupe spécial linéaire d'un espace vectoriel <i>E</i> de dimension finie sur <i>K</i> est le groupe <i>SL</i>(<i>E</i>) des éléments du <a href="/wiki/Groupe_g%C3%A9n%C3%A9ral_lin%C3%A9aire" title="Groupe général linéaire">groupe général linéaire</a> <i>GL</i>(<i>E</i>) dont le déterminant est égal à 1. </p><p>Cette définition admet différentes généralisations : <a href="/wiki/Groupe_g%C3%A9n%C3%A9ral_lin%C3%A9aire#Groupe_spécial_linéaire" title="Groupe général linéaire">une, immédiate, sur un anneau commutatif</a> et deux variantes sur des <a href="/wiki/Corps_gauche" title="Corps gauche">corps non nécessairement commutatifs</a>, dont l'une sur des <a href="/wiki/Alg%C3%A8bre_centrale_simple" class="mw-redirect" title="Algèbre centrale simple">corps qui sont de dimension finie sur leur centre</a>. </p><p><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 SL_{n}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SL_{n}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b93d4f43a3015ade2ac0c4f61171df4b2e5d1f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.788ex; height:2.843ex;" alt="{\displaystyle SL_{n}(\mathbb {R} )}"></span> est un <a href="/wiki/Groupe_de_Lie" title="Groupe de Lie">groupe de Lie</a> <a href="/wiki/Connexit%C3%A9_(math%C3%A9matiques)" title="Connexité (mathématiques)">connexe</a>, dont l'<a href="/wiki/Alg%C3%A8bre_de_Lie" title="Algèbre de Lie">algèbre de Lie</a> est constituée des matrices de <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 M_{n}(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{n}(\mathbb {R} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3260737a99bf9c22cd56e77a02a9dbe62b8c1609" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.96ex; height:2.843ex;" alt="{\displaystyle M_{n}(\mathbb {R} )}"></span> de <a href="/wiki/Trace_(alg%C3%A8bre)" title="Trace (algèbre)">trace</a> nulle. </p> <div class="mw-heading mw-heading2"><h2 id="Article_connexe">Article connexe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Groupe_sp%C3%A9cial_lin%C3%A9aire&veaction=edit&section=1" title="Modifier la section : Article connexe" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Groupe_sp%C3%A9cial_lin%C3%A9aire&action=edit&section=1" title="Modifier le code source de la section : Article connexe"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Groupe_de_Steinberg_(K-th%C3%A9orie)" title="Groupe de Steinberg (K-théorie)">Groupe de Steinberg (K-théorie)</a></li> <li><a href="/wiki/D%C3%A9composition_d%27Iwasawa" title="Décomposition d'Iwasawa">Décomposition d'Iwasawa</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail de l’algèbre"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/24px-Arithmetic_symbols.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/36px-Arithmetic_symbols.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a3/Arithmetic_symbols.svg/48px-Arithmetic_symbols.svg.png 2x" data-file-width="210" data-file-height="210" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Alg%C3%A8bre" title="Portail:Algèbre">Portail de l’algèbre</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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