子群c─正规性对群结构的影响

THE INFLUENCE OF SUBGROUP c─NORMALITY OF SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

群 G的一个子群 H称为在 G中 c正规 ,如果存在一个正规子群 K ,使得 G=H K且 H∩ G≤ HG,其中 HG=Core G( H ) =∩ x∈ GHx 是包含在 H中的 G的最大正规子群 .该文利用子群 c正规性给出一个群为可解群的一些条件 ,主要定理有 :1 )设 G为群 ,若存在 P∈ Syl2 ( G) ,P为 c 正规于 G,则 G可解 ;2 )设N为群 G的非单位正规子群 ,则 N可解当且仅当 G的任意不包含 N的极大子群 M为 c正规于

A subgroup H of a finite group G is said to be cnormal in G if there exists a normal subgroup K of G such that G=HK and H∩G≤H G, where H G=Core G(H)=∩ x∈GH x is the largest normal subgroup of G that is contained in H. In this paper, the authors give some conditions of a soluble group by using cnormality of subgroups and get the main theorems: 1) Let G be a finite group. If there exists a Sylow 2subgroup P of G such that P is cnormal in G, then G is solvable; 2) Let N be a nontrivial normal subgroup of a group G. Then N is solvable if and only if every maximal subgroup of G not containing N is cnormal in G.