2-极大子群次正规的有限群的一个注记

A Note on Subnormal Finite Groups with 2-Maximal Subgroups

设H是有限群G的一个子群,若存在G的极大子群K,使得H是K的极大子群,则称H为G的一个2-极大子群.本文考查了群G的所有2-极大子群均在G中次正规时对有限群G结构的影响,得到内幂零群为超可解群的两个充分条件;当G的F rattin i子群为1时,考虑F(G)的所有极小子群均在G中正规及群G阶的素因子之间的关系,得到群G幂零的一个充分条件.

A subgroup H of a finite group G is said to be 2-maximal subgroup in G if there is a subgroup K which is a maximal subgroup of G such that H is a maximal subgroup of K. After taking into account of the impact of all the subnormal 2-maximal subgroups in G on the structure of the finite group G, the author has obtained two sufficient conditions for finite minimal non-nilpotent groups to be super-solvable, and obtained a sufficient condition for finite groups to be nilpotent by taking into account of the fact that every minimal subgroup of F（G） is normal in G and of the relationship of the prime divisors of ｜G｜ while Ф（G）=1.