极大子群的性质对有限群结构的影响

设H为有限群G的一个子群.称H在G中是s-半正规的,若对任意的素数p||G|,只要(p,|H|)=1,就有PH=HP,其中P∈Syl_p(G);称H在G中是c-可补的,若存在G的子群N,使得G=HN且H∩N≤H_G=Core_G(H).证明了下面定理 设F是一个包含超可解群类U的饱和群系,H△G,且G/H∈F.则G∈F,若下列条件之一成立:(1)若H的每个Sylow子群的所有极大子群在G中或者s-半正规或者c-可补;(2)若F~*(H)的每个Sylow子群的所有极大子群在G中或者s-半正规或者c-可补,其中F~*(H)是H的广义Fitting子群.该定理统一了最近的一些结果.

The Influence of the Properties of Maximal Subgroups on the Structure of a Finite Group

A subgroup H of a finite group G is called s-seminormal in G if it permuts with every Sylow p-subgroup of G with(p,|H|)=1;a subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup N of G such that and H∩N≤H_G=Core_G(H).In this paper we prove following:Theorem Let F be a saturated formation containing U,the class of all supersolvable groups and G a group with a normal subgroup H such that G/H∈F.Then G∈F if one of following holds:(1)all maximal subgroups of any Sylow subgroup of H are either s-seminormal or c-supplemented in;(2)all maximal subgroups of all Sylow subgroups of F~*(H),the generalized Fitting Subgroup of H,are either s-seminormal or c-supplemented in G.This unifies some recent results.