有限群的最大子群的性质对群结构的影响

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.

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

A subgroup of a finite group G is called π-quasinormal in G if it permutes with every Sylow subgroup of G. A subgroup H of a group G is said to be c-supplemented in G if there exists a subgroup N of G such that G = HN and H∩N ≤ HG = CoreG(H). In this paper we prove: Let F be a saturated formation containing u, the class of all supersolvable groups and suppose that G is 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 all Sylow subgroups of H are either π-quasinormal or c-supplemented in G; (2) All maximal subgroups of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, are either π-quasinormal or csupplemented in G. These unify some recent results.