有限群超可解的几个充分条件

Some Sufficient Conditions for Supersolubility of Groups

利用弱拟正规子群概念,经推导得到有限群超可解的几个充分条件.若有限群G满足下列任意一个条件:(1)G的一个极大且循环的子群M在G中弱拟正规;(2)G的一个具有素数幂指数的子群M的Sylow子群及它的Sylow子群的极大子群在G中弱拟正规;(3)G可解,且G的一个极大正规子群M的极大子群在G中弱拟正规;(4)G可解,G的Sylow子群的循环子群在G中弱拟正规;(5)G可解,G的Sylow子群的极大子群在G中弱拟正规;(6)G=AB,A∈Hallπ(G),B∈Hallπ′(G),且A与B的Sylow子群均在G中弱拟正规,则G超可解.

The main purpose of the present paper is to prove the following some theorems： for a finite groupG , if G satisfies one of the following conditions, then G is supersolvable; （1） A maximal and cyclic subgroup of G is weakly quasi-normal in G. （2） All Sylow subgroups and all maximal subgroups of Sylow subgroups of a subgroup which has index power of a prime of G are weakly soluble quasi-normal in G. （3） All maximal subgroups of a maximal subgroup of G which is are weakly quasi-normal in G. （4） All cyclic subgroups of Sylow subgroups of G which is soluble are weakly quasi-normal in G. （5） All maximal subgroups of is soluble are weakly quasi-normal in G. （6） Let G be the products of A which is a Hall - π subgroup of G and B a Hall -π＇ subgroup. All Sylow subgroups of A and B are weakly quasi- normal in G.