弱Φ-可补子群对p-幂零群结构的影响

The Influence of Weakly Φ-supplemented Subgroups on the Structure of p-nilpotent Groups

设H是有限群G的一个子群,H在G中是弱Φ-可补的,如果存在G的一个子群K,使得G=HK且H∩K≤Φ(H),其中Φ(H)是H的Frattini子群.利用p阶和p^2阶子群的弱Φ-可补性,得到如下结论:1)设G是有限群,p是|G|的满足(|G|,p-1)=1的素因数.设E是G的一个正规子群使得G/E是p-幂零群.若E p∩G N p的每个阶为p或4循环子群均在G中弱Φ-可补,那么G是p-幂零群.2)设G有限群,p是|G|满足(|G|,p^2-1)=1的素因数.设E是G的正规子群使得G/E是p-幂零的.若E p∩G N p的每个阶为p^2的子群均在G中弱Φ-可补,则G是p-幂零的.由这些结论,得到了一系列推论,推广了已知结果.

A subgroup H of G is said to be weakly Φ -supplemented in G , if there exists a subgroup K of G , such that G=HK and H∩K≤Φ(H) . In this paper, we investigate further the influence of weakly Φ -supplemented subgroups of order of p or p^2 on the p -nilpotent groups. The following results are obtained: 1) Let G be a finite group, E be a normal subgroup of G such that G/E is p -nilpotent.If every cyclic subgroup of E p ∩G N p of order p or 4 is weakly Φ -supplemented in G , then G is p -nilpotent. 2) Let G be a finite group, p be a prime divisor of |G| such that (|G|,p^2 -1) =1. Let E a normal subgroup of G such that G/E is p -nilpotent. If every subgroup of E p ∩G N p of order p^2 is weakly Φ -supplemented in G , then G is p -nilpotent. By these results, we may get a series of corollaries, which contain known results.