{"id":178,"date":"2024-03-24T12:00:00","date_gmt":"2024-03-24T11:00:00","guid":{"rendered":"https:\/\/www.maths-sup.com\/?p=178"},"modified":"2024-03-25T00:09:03","modified_gmt":"2024-03-24T23:09:03","slug":"sous-groupes-compacts-de-glnr","status":"publish","type":"post","link":"https:\/\/www.maths-sup.com\/index.php\/2024\/03\/24\/sous-groupes-compacts-de-glnr\/","title":{"rendered":"Sous-groupes compacts de GLn(R)"},"content":{"rendered":"\n<p>Pour <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> un entier, on note:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{GL}_n(\\mathbb{R})<\/span> le <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Groupe_g%C3%A9n%C3%A9ral_lin%C3%A9aire\">groupe lin\u00e9aire<\/a> d&rsquo;ordre <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>.<\/li>\n\n\n\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{O}_n(\\mathbb{R})<\/span> le <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Groupe_orthogonal\">sous-groupe orthogonal<\/a> d&rsquo;ordre <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span>.<\/li>\n<\/ul>\n\n\n\n<p>On sait que <span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{O}_n(\\mathbb{R}<\/span> est un groupe compact (car ferm\u00e9 et born\u00e9e dans l&rsquo;espace vectoriel des matrices).<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Le th\u00e9or\u00e8me<\/h2>\n\n\n\n<p><strong>Th\u00e9or\u00e8me:<\/strong><br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> un entier.<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">K \\subset \\mathrm{GL}_n(\\mathbb{R})<\/span> un sous-groupe compact.<br>Alors: il existe <span class=\"katex-eq\" data-katex-display=\"false\">P \\in \\mathrm{GL}_n(\\mathbb{R})<\/span> tel que<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">P \\cdot K \\cdot P^{-1} \\subset \\mathrm{O}_n(\\mathbb{R}).<\/span><\/p>\n\n\n\n<p>Le th\u00e9or\u00e8me s&rsquo;\u00e9nonce de mani\u00e8re plus concise: tout sous-groupe compact de <span class=\"katex-eq\" data-katex-display=\"false\">K \\subset \\mathrm{GL}_n(\\mathbb{R})<\/span> est conjugu\u00e9 \u00e0 un sous-groupe <span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{O}_n(\\mathbb{R}<\/span>.<\/p>\n\n\n\n<p>Le fait que <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> soit conjugu\u00e9 \u00e0 un sous-groupe de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathrm{O}_n(\\mathbb{R}<\/span> s&rsquo;interpr\u00e8te par le fait qu&rsquo;il existe un produit scalaire de <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}|^n<\/span> invariant par tout \u00e9l\u00e9ment de <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Preuve du th\u00e9or\u00e8me<\/h2>\n\n\n\n<p>On note <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> l&rsquo;espace vectoriel des formes quadratiques sur <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}^n<\/span>.<\/p>\n\n\n\n<p>On d\u00e9finit le morphisme de groupe continue <span class=\"katex-eq\" data-katex-display=\"false\">\\phi : \\mathrm{GL}_n(\\mathbb{R}) \\rightarrow \\mathrm{GL}(Q) <\/span> par<\/p>\n\n\n\n<span class=\"katex-eq\" data-katex-display=\"false\">\\phi(g)(q) = q \\circ g^{-1}<\/span>\n\n\n\n<p><strong>Le but est de montrer<\/strong> qu&rsquo;il existe <span class=\"katex-eq\" data-katex-display=\"false\">q_0 \\in Q<\/span> d\u00e9finie positive telle que:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall g \\in K, \\phi(g)(q_0) = q_0[.\/katex]&lt;\/p&gt;\n\n\n\n&lt;p&gt;On note [katex]K' \\subset \\mathrm{GL}(Q)<\/span> l'image de <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span>.<br>Comme <span class=\"katex-eq\" data-katex-display=\"false\">K<\/span> est compact et <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span> est continue, <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> <strong>est compact<\/strong>.<\/p>\n\n\n\n<p>On se fixe <span class=\"katex-eq\" data-katex-display=\"false\">q_0 \\in Q<\/span> d\u00e9finie positive.<br>On note <span class=\"katex-eq\" data-katex-display=\"false\">X \\subset Q<\/span> le sous-ensemble:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">X := \\{ g \\cdot q_0 ~,~ g \\in G \\} <\/span><\/p>\n\n\n\n<p>Alors:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est compact, car c'est l'image du compact <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> par l'application continue <span class=\"katex-eq\" data-katex-display=\"false\">g \\mapsto g \\cdot q_0<\/span>.<\/li>\n\n\n\n<li>Tout \u00e9l\u00e9ment de <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est d\u00e9finie positive.<\/li>\n\n\n\n<li><span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> est stable par l'action de <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span><\/li>\n<\/ul>\n\n\n\n<p>On note <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> l'enveloppe convexe de <span class=\"katex-eq\" data-katex-display=\"false\">X<\/span> dans <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span>.<br>Alors:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est compact d'apr\u00e8s le <a href=\"https:\/\/fr.wikipedia.org\/wiki\/Th%C3%A9or%C3%A8me_de_Carath%C3%A9odory_(g%C3%A9om%C3%A9trie)\">th\u00e9or\u00e8me de Carath\u00e9odorie<\/a>.<\/li>\n\n\n\n<li>Tout \u00e9l\u00e9ment de <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est d\u00e9finie positive (car un barycentre \u00e0 coefficients positifs de formes quadratiques d\u00e9finies positives reste d\u00e9finie positive).<\/li>\n\n\n\n<li><span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est stable par l'action de <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>.<\/li>\n<\/ul>\n\n\n\n<p>Le lemme suivant montre qu'il existe <span class=\"katex-eq\" data-katex-display=\"false\">q \\in C<\/span> fixe par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>.<br>CQFD.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Un lemme de point fixe<\/h2>\n\n\n\n<p>Soit <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span> un <span class=\"katex-eq\" data-katex-display=\"false\">\\mathbb{R}<\/span>-espace vectoriel de dimension finie.<\/p>\n\n\n\n<p><strong>Lemme:<\/strong><br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">G \\subset \\mathrm{GL}(V)<\/span> un <strong>sous-groupe compact<\/strong>.<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">C \\subset V<\/span> un <strong>convexe compact stable par<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, i.e <span class=\"katex-eq\" data-katex-display=\"false\">G \\cdot C \\subset C<\/span>.<br>Alors il existe <span class=\"katex-eq\" data-katex-display=\"false\">c_0 \\in C<\/span> fixe par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, i.e.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall g \\in G, \\quad g(c_0) = c_0<\/span><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Preuve du lemme<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Etape 1<\/h3>\n\n\n\n<p>Soit <span class=\"katex-eq\" data-katex-display=\"false\">g \\in \\mathrm{End}(V)<\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">g(C)<\/span>.<br><strong>Montrons que<\/strong>:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\exists v \\in C, \\forall g \\in G, \\quad g(v) = v<\/span>.<\/p>\n\n\n\n<p>On se fixe <span class=\"katex-eq\" data-katex-display=\"false\">c_0<\/span>.<br>Pour tout <span class=\"katex-eq\" data-katex-display=\"false\">n \\in \\mathbb{N}<\/span> on note:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">c_n := g^n(u_0) = g \\circ g \\circ \\cdots \\circ g(u_0)<\/span>.<\/li>\n\n\n\n<li><span class=\"katex-eq\" data-katex-display=\"false\">u_n := \\tfrac{1}{n} \\sum_{i=0}^{n-1} c_i<\/span> (c'est la moyenne de Cesaro de <span class=\"katex-eq\" data-katex-display=\"false\">(c_n)<\/span>).<\/li>\n<\/ul>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est <strong>convexe et stable par<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span>, <br>les <span class=\"katex-eq\" data-katex-display=\"false\">c_n<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">u_n<\/span> sont dans <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est <strong>compact<\/strong>, il existe une extraction <span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span>,<br>tel que <span class=\"katex-eq\" data-katex-display=\"false\">(u_{\\phi(n)})_n<\/span> converge dans <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span>.<br>On note <span class=\"katex-eq\" data-katex-display=\"false\">v \\in C<\/span> sa limite.<\/p>\n\n\n\n<p>Pour tout <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> on a:<\/p>\n\n\n\n<span class=\"katex-eq\" data-katex-display=\"false\">g(u_n) - u_n = \\tfrac{1}{n} \\sum_{i=0}^{n-1} u_{i+1} - u_i = \\tfrac{1}{n}(u_n - u_0)<\/span>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est born\u00e9, on a: <span class=\"katex-eq\" data-katex-display=\"false\">\\lim_{n \\to \\infty} \\|g(u_n) - u_n\\| = 0<\/span>.<\/p>\n\n\n\n<p>En passant \u00e0 la limite sur <span class=\"katex-eq\" data-katex-display=\"false\">(\\phi(n))_n<\/span>, <strong>on en d\u00e9duit que<\/strong>: <span class=\"katex-eq\" data-katex-display=\"false\">g(v) = v<\/span>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Etape 2<\/h3>\n\n\n\n<p>On se fixe <span class=\"katex-eq\" data-katex-display=\"false\">\\| \\cdot \\|_2<\/span> une norme euclidienne quelconque sur <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span>.<br>On d\u00e9finit le norme suivante sur <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall x \\in V,  N(x) := \\max_{g \\in G}(\\| g(x)\\|_2)<\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> est compact, le <span class=\"katex-eq\" data-katex-display=\"false\">\\max<\/span>est bien d\u00e9fini.<\/p>\n\n\n\n<p><strong>Montrons que<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> est invariant par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, i.e.:<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall x \\in V, \\forall g \\in G, \\quad N(g(x)) = N(x)<\/span><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> est strictement convexe, i.e.:<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\forall x,y \\in V,  N(x+y) = N(x) + N(y) \\longrightarrow (\\exists \\lambda \\geq 0, x = \\lambda ~ y)<\/span><\/p>\n\n\n\n<p>Le premier point, vient du fait que <span class=\"katex-eq\" data-katex-display=\"false\">g \\dot G = G<\/span>, car <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span> est un sous-groupe<\/p>\n\n\n\n<p>Pour le deuxi\u00e8me point: soit <span class=\"katex-eq\" data-katex-display=\"false\">x,y \\in V<\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">N(x+y) = N(x) + N(y)<\/span>.<br>Soit <span class=\"katex-eq\" data-katex-display=\"false\">g_0 \\in G<\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">N(x+y) = \\| g_0(x+y) \\|_2<\/span>.<br>On a alors<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">N(x+y) = \\|g_0(x) + g_0(y) \\|_2 \\leq \\|g_0(x) \\|_2 + \\|g_0(y) \\|_2 \\leq N(x) + N(y)<\/span><\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">N(x+y) = N(x) + N(y)<\/span>, on a donc <span class=\"katex-eq\" data-katex-display=\"false\">\\|g_0(x) + g_0(y) \\|_2 = \\|g_0(x) \\|_2 + \\|g_0(y) \\|_2 <\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">\\| \\cdot \\|_2<\/span> est <strong>strictement convexe<\/strong>,<br>il existe <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda \\geq 0<\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">g_0(x) = \\lambda ~ g_0(y)<\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">g_0<\/span> est <strong>lin\u00e9aire injectif<\/strong>, on a <span class=\"katex-eq\" data-katex-display=\"false\">x = \\lambda ~ y<\/span>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Etape 3<\/h3>\n\n\n\n<p>Soit <span class=\"katex-eq\" data-katex-display=\"false\">g_1, g_2, \\dots, g_p \\in G<\/span> une famille finie de <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>.<br><strong>Montrons que<\/strong><\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">\\exists v \\in C, \\forall i=1 \\dots p, \\quad g_i(v) = v<\/span>.<\/p>\n\n\n\n<p>On pose <span class=\"katex-eq\" data-katex-display=\"false\">g := \\tfrac{1}{p}(g_1 + \\cdots + g_p)  \\in \\mathrm{End}(V)<\/span> la moyenne des <span class=\"katex-eq\" data-katex-display=\"false\">g_i<\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est convexe, <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est stable par <span class=\"katex-eq\" data-katex-display=\"false\">g<\/span>.<\/p>\n\n\n\n<p><strong>D'apr\u00e8s l'\u00e9tape 1<\/strong>, il existe <span class=\"katex-eq\" data-katex-display=\"false\">v \\in C<\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">g(v) = v<\/span>.<\/p>\n\n\n\n<p>On a<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">N(g(v)) =\\frac{1}{p} N(g_1(v) + \\cdots g_p(v)) \\leq \\tfrac{1}{p}(N(g_1(v)) + \\cdots + N(g_p(v)))<\/span><\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> est invariant par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>, <span class=\"katex-eq\" data-katex-display=\"false\">N(g_i(v)) = N(v)<\/span>, et le membre de droite vaut donc <span class=\"katex-eq\" data-katex-display=\"false\">N(v)<\/span>.<br>Comme <span class=\"katex-eq\" data-katex-display=\"false\">g(v) = v<\/span>, le membre de gauche vaut <span class=\"katex-eq\" data-katex-display=\"false\">N(v)<\/span>.<br>Donc <strong>on a \u00e9galit\u00e9 au milieu<\/strong>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> <strong>est strictement convexe<\/strong>, les <span class=\"katex-eq\" data-katex-display=\"false\">(g_i(v))_{i=1 \\cdots p}<\/span> <strong>sont positivement li\u00e9s<\/strong>.<br>Comme ils sont aussi de m\u00eame norme, ils sont tous \u00e9gaux \u00e0 leur moyenne qui vaut <span class=\"katex-eq\" data-katex-display=\"false\">g(v) = v<\/span>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Fin de la preuve du lemme<\/h3>\n\n\n\n<p>Pour tout <span class=\"katex-eq\" data-katex-display=\"false\">g \\in G<\/span> on note:<\/p>\n\n\n\n<p class=\"has-text-align-center\"><span class=\"katex-eq\" data-katex-display=\"false\">F_g := \\{ c \\in C ~|~ g(c) = c \\}<\/span>.<\/p>\n\n\n\n<p>Comme <span class=\"katex-eq\" data-katex-display=\"false\">c \\mapsto g(c)-c <\/span> est continue,<br><span class=\"katex-eq\" data-katex-display=\"false\">F_g<\/span> <strong>est un ferm\u00e9 de<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span>.<\/p>\n\n\n\n<p><strong>Montrons que<\/strong> <span class=\"katex-eq\" data-katex-display=\"false\">\\bigcap_{g \\in G} F_g<\/span> <strong>est non vide<\/strong>.<\/p>\n\n\n\n<p>On raisonne par l'absurde.<br>Supposons que l'intersection est vide.<br>Comme <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> est compact,<br>il existe une famille finie <span class=\"katex-eq\" data-katex-display=\"false\">g_1, \\dots, g_p<\/span>,<br>telle que <span class=\"katex-eq\" data-katex-display=\"false\">\\bigcap_{i=1 \\dots p} F_{g_i} = \\empty<\/span>.<br><strong>Ce qui contredit l'\u00e9tape 3.<\/strong><\/p>\n\n\n\n<p>Donc il existe <span class=\"katex-eq\" data-katex-display=\"false\">v \\in \\bigcap_{g \\in G} F_g<\/span>, qui est donc un point fixe par <span class=\"katex-eq\" data-katex-display=\"false\">G<\/span>.<br>CQFD.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Le\u00e7ons concern\u00e9es<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>106 - Groupe lin\u0013\u00e9aire d'un espace vectoriel de dimension fi\fnie E, sous-groupes de GL(E). Applications.<\/li>\n\n\n\n<li>119 - Exemples d'actions de groupes sur les espaces de matrices.<\/li>\n\n\n\n<li>121 - Matrices \u0013\u00e9quivalentes. Matrices semblables. Applications.<\/li>\n\n\n\n<li>131 - Formes quadratiques sur un espace vectoriel de dimension fi\fnie. Orthogonalit\u00e9e, isotropie. Applications.<\/li>\n\n\n\n<li>135 - Isom\u0013\u00e9tries d'un espace affi\u000ene euclidien de dimension \ffinie. Forme r\u00e9duite. Applications en dimensions 2 et 3.<\/li>\n\n\n\n<li>137 - Barycentres dans un espace a\u000efine r\u0013\u00e9el de dimension fi\fnie; convexit\u00e9\u0013. Applications.<\/li>\n\n\n\n<li>141 - Utilisation des groupes en g\u0013\u00e9om\u0013\u00e9trie.<\/li>\n\n\n\n<li>203 - Utilisation de la notion de compacit\u0013\u00e9.<\/li>\n\n\n\n<li>206 - Th\u0013\u00e9or\u00e8mes de point fi\fxe, exemples et applications.<\/li>\n<\/ul>\n\n\n\n<p>Si tu as des remarques ou des questions, alors n'h\u00e9site pas \u00e0 \u00e9crire en commentaire.<\/p>\n\n\n\n<p><br><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pour un entier, on note: On sait que est un groupe compact (car ferm\u00e9 et born\u00e9e dans l&rsquo;espace vectoriel des matrices). Le th\u00e9or\u00e8me Th\u00e9or\u00e8me:Soit un entier.Soit un sous-groupe compact.Alors: il existe tel que Le th\u00e9or\u00e8me s&rsquo;\u00e9nonce de mani\u00e8re plus concise: tout sous-groupe compact de est conjugu\u00e9 \u00e0 un sous-groupe . [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":252,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[11,12,23,10,24,16,17,18,19,20,21,14,22],"tags":[],"class_list":["post-178","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-action-de-groupe","category-compacite","category-convexite","category-groupe","category-groupe-lineaire","category-lecon-106-2","category-lecon-119","category-lecon-121","category-lecon-131","category-lecon-135","category-lecon-137","category-lecon-203","category-lecon141"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.0 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Sous-groupes compacts de GLn(R) - Maths-Sup<\/title>\n<meta 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