﻿ Π-可补充子群对有限群结构的影响

# Π-可补充子群对有限群结构的影响The Influence of Π-Supplemented Subgroups on Finite Groups

H是群G的子群，如果对群G的任意主因子L/K，|G/K:NG/K(HK/KIL/K)|是 π(HK/KIL/K)-数，则称H在G中满足π-性质；如果存在G的子群T使得G=HT，并且HIT≤I≤H，其中I在G中满足π-性质，则称H在G中是π-可补充的。利用子群的π-可补充性，得到了有限群p-幂零性的一个新的判定方法。

Abstract: Let H be a subgroup G . If every chief factor L/K of G, |G/K:NG/K(HK/KIL/K)| is a π(HK/KIL/K)-number, then H is called satisfying π-property in G. If there exists a subgroup  T of  G such that  G=HT and HIT≤I≤H , where I satisfies π-property in G , then H is called  π-supplemented in G . By the property of  π-supplemented, some new criterion of p-nilpotency of finite groups is obtained.

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