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ISBN10: 2705614486
ISBN13: 9782705614485
Publisher: Hermann
Published: Sep 15 2019
Pages: 172
Weight: 0.62
Height: 0.37 Width: 6.69 Depth: 9.61
Language: French
ISBN13: 9782705614485
Publisher: Hermann
Published: Sep 15 2019
Pages: 172
Weight: 0.62
Height: 0.37 Width: 6.69 Depth: 9.61
Language: French
Les notions classiques sont abordées de façon élémentaire et rigoureuse: groupe, sous-groupe, sous-groupe distingué, groupe quotient, groupe cyclique, produit direct, somme restreinte, somme directe de groupes, produit semi-direct de groupes, groupe opérant sur un ensemble, groupe libre, produit libre de groupes, groupe défini par générateurs et relations. Les résultats de base sont démontrés: structure du groupe alterné An, célèbres théorèmes de Sylow, suite dérivée, suite centrale descendante en relation avec les groupes résolubles ou nilpotents à la structure des groupes abéliens de type fini est complètement élucidée par des opérations élémentaires sur les lignes et colonnes d'une matrice. Le dernier paragraphe, le plus riche et le plus original, regroupe 75 exercices dont certains, faciles, permettent de se familiariser avec des groupes finis d'ordre petit . D'autres exercices rencontrent des groupes célèbres provenant de la géométrie, de l'algèbre linéaire et de l'arithmétique, comme les groupes de polyèdres réguliers en dimension 3, les fameux groupes de Coxeter, le groupe fondamental d'une surface de Riemann, les célèbres théorèmes de Schur, Burnside et Jordan sur les sous-groupes finis de Gln(C). Cet ouvrage s'adresse aussi bien aux étudiants de premier cycle que de licence ou de maîtrise de mathématiques. C'est également un outil particulièrement utile pour les candidats aux concours du CAPES et de l'agrégation.
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