有关C-可补子群

On the C-supplement Subgroups

关键词:  群系  有限群  C-可补

Key words:  formations  finite groups  C-supplement

摘要:

运用群系理论讨论Sylow子群的极大子群和Sylow子群的二次极大子群,以及极小子群对有限群结构的影响.得到:(1)设G是与A₄无关的有限群,P是|G|的最小素因数,F是包含N_p的群系,则G∈F的充要条件为G存在一个正规子群,使得G/H∈F且H的Sylow p-子群的二次极大子群在G中C-可补;(2)设F是非空子群闭的局部群系,G是有限群,p是|G|的最小素因数且G^F是可解,那么G∈F⇔G存在正规子群N使得G/N∈F且对于P∈Syl_p(N),P∩G^F的2²阶循环子群在G中C-可补且极小子群皆包含在Z_∞^F(G)中.

Abstract:

The C-supplement condition on the maximal subgroup or the second maximal subgroup of Sylow subgroup and the minimal subgroup of G are used to study the structure of G by the theory of formations. The following results are obtained. (l) Let G be a finite group which is A₄-free and G be a formation containing N_p, where p is the smallest prime number dividing|G|. Then G∈F, and there exists a normal subgroup H of G such that G/N∈F and the second maximal subgroup of Sylow p- subgroup of H is C-supplement in G. (2) Let F be non-empty and subgroup-closed formation, G be a finite group and G^F is solvable. Then G∈F,and there exists a normal subgroup N of G such that G/N∈F,and for p∈Syl_p(G), the subgroups of prime order and the subgroup of order 4 are contained in Z_∞^F(G). The result above generalizes some known ones.