By Karlheinz Spindler

ISBN-10: 0824791444

ISBN-13: 9780824791445

A accomplished presentation of summary algebra and an in-depth remedy of the functions of algebraic options and the connection of algebra to different disciplines, akin to quantity conception, combinatorics, geometry, topology, differential equations, and Markov chains.

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XII, th. ] (b) Soit 1 --* P --~ E --* G --* 1 une extension de G par un pro-p-groupe P. Montrer que tout rel~vement de N dans E se prolonge en un rel~vement de G. ] 4) Donner un exemple d'extension 1 --* P --* E --* G --, 1 de groupes profinis aymat les propri@t6s suivantes: (i) P est un pro-p-groupe. (ii) G est fini. (iii) Un p-groupe de Sylow de G se relive dams E. 5. M o d u l e dualisant Soit G u n groupe profini. Nous noterons C~ (resp. C~) la cat@gorie des Gmodules discrets A qui sont des groupes finis (resp.

Ils sont donc de dimension cohomologique _ 1. On va pr6ciser un peu le corollaire 1 dans le cas particulier off I e s t fini. Si gl, . . , gn sont des ~l~ments de G, nous dirons que les gi engendrent G (topologiquement) si le sous-groupe qu'ils engendrent (all sens alg~brique) est dense dans G; il revient au m~me de dire que tout quotient G / U , avec U ouvert, est engendrd par les images des 9i. P r o p o s i t i o n 25. Soient gl, . . , gn des dldments d'un pro-p-groupe G. Les conditions suivantes sont dquivalentes: (a) gl, .

A) Soit H u n sous-groupe ferm4 de G, distinct de G. Montrer que R e s : gn(G) ~ Hn(H) est 0. [Se r a m e n e r au cas off H est ouvert, et utiliser la partie (4) de la d 4 m o n s t r a t i o n de la prop. e. que H n'est pas ouvert. Montrer que cd(H)

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Abstract Algebra with Applications by Karlheinz Spindler

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