Fusion of π-subgroups, isometries, and normal complements
Fusion of π-subgroups, isometries, and normal complements
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π 子群、等距和正态补集的融合
DOI：
10.1016/0021-8693(87)90121-9
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发表时间：
1987
影响因子：
0.9
通讯作者：
G. Robinson
中科院分区：
文献类型：
--
作者：
G. Robinson
本文的目的是隔离一些群论的要求，这将确保有限群G的子群H将控制其某些元素的共轭性，以这种方式，H的广义特征将扩展到G的广义特征。使用的方法并不难。当然，Alperin的融合定理起着很大的作用。在最粗略的层次上，阿尔佩林的融合定理说，如果H包含G的一个Sylow p-子群，并且N（Q）= C，（Q）NH（Q），只要Q是H的一个非平凡p-子群，那么H控制它的p-元的共轭性。我们指出，当试图控制共轭的rc-元素的子群，H（rc一组素数），知识的Brauer基本x-子群的自动化是很重要的。众所周知，控制一个Hall子群的元素的共轭性通常不足以构造该子群的正规补（例如，当p 2 5是素数时，S1-I是S的Hall子群，它控制其元素的共轭性，但在S1中没有S1_ r的正规补）。我们试图隔离额外的条件，完全由局部子群的结构决定，这将确保相对正规补的存在。
The aim of this paper is to isolate some group-theoretic requirements which will ensure that a subgroup, H, of the finite group G will control the conjugacy of certain of its elements, in such a manner that generalized characters of H will extend to generalized characters of G. The methods used are not difficult. Of course, Alperin’s fusion theorem plays a large role. At the very crudest level, Alperin’s fusion theorem says that if H contains a Sylow p-subgroup of G, and N&Q)= C,(Q) NH (Q) whenever Q is a nontrivial p-subgroup of H, then H controls the conjugacy of its p-elements. We point out that when attempting to control conjugacy of rc-elements of a subgroup, H (rc a set of primes), a knowledge of the automizers of Brauer elementary x-subgroups of H is important. It is well known that control of conjugacy of elements of a Hall subgroup is not generally sufficient to construct a normal-complement to that subgroup (eg, when p 2 5 is a prime, S,-I is a Hall subgroup of S, which controls the conjugacy of its elements, yet there is no normal complement to S, _ r in S,). We try to isolate extra conditions, determined completely by the structure of local subgroups, which will ensure the existence of relative normal complements.