有限群的可解性

The Solvability of Finite Groups

C-正规子群第一次被提出并被用来讨论了有限群的结构,之后得到人们的广泛关注。我们利用C-正规子群对有限群的可解性进行了讨论,得到了可解群的一些新的充分条件。主要结果有(1)设G是有限群,H是G的偶阶幂零Hall子群,M是H的极大子群,若M的2-sylow子群在G中C-正规,则G是可解群;(2)设M是G的指数为2的偶阶极大子群,若M是内幂零群,且M的p′-sylow子群在G中C-正规,则G可解;(3)设H是G的π-Hall子群,且2∈π,若H幂零且H的某个极大子群M在G中C-正规,则G是可解群。

The C-normal subgroup was firstly presented and used to discuss the structure of finite group.In this paper,we use the C-normal subgroup to discuss the solvability of finite group and obtain some new sufficient conditions.Main results are:Theorem 1 let G be a finite group and H be a even nilpotent Hall subgroup,M<H,if M_2∈sly_2(M) is C-normal in G,then G is solvable.Theorem 2 Let G be a finite group and M be a even maximum subgroup, which index is 2 in G,if M is a inner-nilpotent group and M_p∈sly_p(M) is C-normal in G,then G is solvable.Theorem 3 let H be a π-Hall nilpotent subgroup of G,2∈π.If a maximum subgroup of H is C-normal in G,then G is solvable.