{"id":234000,"date":"2022-04-28T17:07:45","date_gmt":"2022-04-28T15:07:45","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=234000"},"modified":"2025-10-25T11:44:32","modified_gmt":"2025-10-25T09:44:32","slug":"structures-algebriques-exercices-corriges","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/structures-algebriques-exercices-corriges\/","title":{"rendered":"Structures alg\u00e9briques exercices corrig\u00e9s"},"content":{"rendered":"\n

Tu connais ton cours sur les structures alg\u00e9briques<\/strong> ? Tu cherches d\u00e9sormais des exercices corrig\u00e9s <\/strong>pour t’entra\u00eener et v\u00e9rifier que tu as bien compris cette notion de math\u00e9matiques ? Tu trouveras tout ce dont tu as besoin juste ici, continue de lire !<\/p>\n\n\n\n

Aborde les structures alg\u00e9briques avec assurance et ma\u00eetrise, en t’appuyant sur l’expertise de nos cours de soutien en algebre<\/a>.<\/strong> \ud83d\udcda<\/p>\n\n\n\n

Exercice d’application sur les structures alg\u00e9briques :<\/strong><\/p>\n\n\n\n

\u23f0 Dur\u00e9e : 20 min<\/p>\n\n\n\n

\ud83d\udcaa Difficult\u00e9 : 1\/3<\/p>\n\n\n\n\nSoit \"(G,\cdot)\" un groupe. Pour tout \"y \in\ G\", on pose \"\tau_y : x \in\ G \mapsto yxy^{-1}\".\n\n\n\n

<\/div>\n\n\n\n\n1. Que dire de \"\tau_y\" lorsque G est ab\u00e9lien ?\n\n\n\n
<\/div>\n\n\n\n\n2. Montrer que pour tout \"y \in\ G\", \"\tau_y\" est un automorphisme de groupes.\n\n\n\n
<\/div>\n\n\n\n\n3. On note \"Int(G) = \{ \tau_y, y \in\ G \}\". Montrer que (\"Int(G),\circ\") est un groupe.\n\n\n\n

Corrig\u00e9 de l’exercice d’application sur les structures alg\u00e9briques<\/strong><\/p>\n\n\n\n\n1. Si \"G\" est ab\u00e9lien, alors \"\tau_y = id_G\".\n\n\n\n

<\/div>\n\n\n\n\n2. \n
  • Soit \"y \in\ G\". Soit \"(x_1,x_2) \in\ G^2\". On a : <\/li>\n\n\n\n
    <\/div>\n\n\n\n\n

      <\/span>   <\/span>\"\[<\/p>\n\n\n\n

    <\/div>\n\n\n\n\nOn a montr\u00e9 que \"\tau_y\" est un morphisme de groupe.\n\n\n\n
    <\/div>\n\n\n\n\n
  • Soit \"y \in\ G\". Soit \"z \in\ G\". On a : <\/li>\n\n\n\n
    <\/div>\n\n\n\n\n

      <\/span>   <\/span>\"\[<\/p>\n\n\n\n

    <\/div>\n\n\n\n\nOn en d\u00e9duit que \"z\" admet un unique ant\u00e9c\u00e9dent par \"\tau_y\" , ainsi \"\tau_y\" est bijective.\n\n\n\n
    <\/div>\n\n\n\n\nOn a montr\u00e9 que \"\tau_y\" est un automorphisme de groupe et \"(\tau_y)^{-1} = \tau_{y^{-1}}\". On a montr\u00e9 que \"Int(G)\" est un sous-groupe de l’ensemble des bijections de \"G\" sur \"G\", donc c’est un groupe.\n\n\n\n
    <\/div>\n\n\n\n\n3. Il est clair que \"Id_G \in\ Int(G)\". Soit \"(y_1,y_2) \in\ G^2\". Montrons que \"\tau_{y_{1}} \circ \tau_{y_{2}} \in\ Aut(G)\". Pour tout \"x \in\ G\", on a :\n

      <\/span>   <\/span>\"\[<\/p>\nAinsi, \"\tau_{y_{1}} \circ \tau_{y_{2}}^{-1} = \tau_{y_1y_2^{-1}}} \in\ Int(G)\".\n\n\n\n

    \n
    \n
    \"livre<\/figure>\n<\/div>\n\n\n\n
    <\/div>\n\n\n\n
    \n
    <\/div>\n\n\n\n

    Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) <\/em>\u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n<\/div>\n<\/div>\n\n\n

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    \n 4\/5 - (6 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

    Tu connais ton cours sur les structures alg\u00e9briques ? Tu cherches d\u00e9sormais des exercices corrig\u00e9s pour t’entra\u00eener et (…)<\/p>\n","protected":false},"author":158,"featured_media":244622,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-234000","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234000","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=234000"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/234000\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/244622"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=234000"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=234000"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=234000"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}