Download Algèbre : Chapitres 1 à 3 by N. Bourbaki PDF

Download Algèbre : Chapitres 1 à 3 by N. Bourbaki PDF

By N. Bourbaki

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Additional resources for Algèbre : Chapitres 1 à 3

Example text

D) H est stable pour la loi de G et la loi de composition induite sur H par la loi de composition de G est une loi de groupe. Il est clair que a) entraîne b). Montrons que b) entraîne a). Il suffit de montrer que H contient l'élément neutre de G. La partie H n'étant pas vide, soit x E H. Alors x-1 E H et e = xx-1 appartient à H. Il est clair que b) entraîne c). Montrons que c) entraîne b). Tout d'abord, H n'étant pas vide possède un élément x. Par suite xx-1 = e est un élément de H. Pour tout élément x de H, x-1 = ex-1 appartient à H; donc les relations x E H, y E H entraînent xy = x(y-1) -1 E H.

4. GROUPES ET GROUPES A OPÉRATEURS 1. Groupes Rappelons la définition suivante (I, p. 15, déf. 6) : DÉFINITION 1. — On appelle groupe un ensemble muni d'une loi de composition associative, possédant un élément neutre et pour laquelle tout élément est inversible. 29 Autrement dit, un groupe est un monade (I, p. 12, déf. 2) dans lequel tout élément est inversible. Une loi de composition sur un ensemble qui y détermine une structure de groupe est appelée une loi de groupe. Si G et H sont deux groupes, un homomorphisme de magmas de G dans H est encore appelé un homomorphisme de groupes.

35, prop. 5, appliquée à Ide, il existe un homomorphisme u de G/N dans G/G' défini par u(xN) = xG' pour tout x E G. Il est immédiat que u est surjectif, de noyau G'/N, d'où l'isomorphisme cherché de (G/N)/(G'/N) sur G/G'. Enfin, c) résulte immédiatement de I, p. 38, prop. 8. 7. Le théorème de Jordan-Helder 9. — On appelle suite de composition d'un groupe à opérateurs G une suite finie (Gi) 0 ,,A,,_„ de sous-groupes stables de G, avec Go = G, G„ = {e} et telle que Gi +1 i n — 1. 40 STRUCTURES ALGÉBRIQUES §4 les quotients de la suite.

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