F-S-可补的子群对有限群结构的影响

The influence of F-S-supplemented subgroups on the structure of finite groups

设F是一个群系.群G的一个子群H在G中F-S-可补,如果存在G的子群K,使得G=HK且K/K∩HG∈F,其中HG表示G包含在H中的最大的正规子群.本文利用群系理论研究子群的F-S-可补性对有限群结构的影响,得到如下结论:设F是子群闭的局部群系,G是有限群且GF是可解的.则G∈F的充要条件是下列条件之一:(1)G存在正规子群N使得G/N∈F且N的极小子群及4阶循环子群(p=2)均在G中F-S-可补.(2)G存在正规子群N使得G/N∈F,N的4阶循环子群在G中有F-S-补且N的极小子群皆包含在Z∞F(G)中.应用这些结论,可以得到一些推论,其中包括已知的相关结果.

Let F be a class of groups.A subgroup H is called Y-S-supplemented in G if there exists a subgroup K of G such that G = HK and K/K ∩ HG ∈Y, where HG is the maximal normal subgroup of G contained in H. In this paper,the F-S-supplemented subgroups of G is used to study the structure of G by the theory of formations. The following results are obtained： Let G be a finite group, F be a formation and GF be soluble. Then G ∈ F if and only if one of the following two conditions： （1） G has a normal subgroup N such that G/N ∈ F and the subgroups of order p or 4 of N are F-S-supplemented in G. （2） G has a normal subgroup N such that G/N ∈f, the subgroups of prime order of N are contained in Z F ∞ （G） and the subgroups of order 4 are F-S-supplemented in G. By these results, we may get a series of corollaries,which contain known results.