有限群的极大子群的正规指数

On the Normal Index of Maximal Subgroups in Finite Groups

利用极大子群的正规指数的概念 ,得到有限群为 p -可解、可解的若干充要条件 .主要证明了如下结果 :设 p是 ｜G｜的最大素因子 ,(1)对任意非幂零的极大子群M∈FG ={M｜M为G的包含Sylow - p子群正规化子的c-极大子群 } ,若G满足下列三个条件之一 :(a)恒有 η(G∶M ) =｜G∶M｜ ;(b)恒有 η(G∶M )无平方因子 ;(c)恒有 η(G∶M )为素数方幂 ;则G是 p-可解的 .(2 )以下命题等价 :①G是可解的 ;②对任意非幂零的极大子群M∈F′G ∩Fp,恒有 η(G∶M ) =｜G∶M｜ ;③对任意非幂零的极大子群M ∈F′G ∩Fp,恒有 η(G∶M )为素数方幂 .

By using the concept of normal index, some necessa ry and sufficient conditions for a finite group to be p-solvable and solvabl e are obtained. In this paper the following results are proved: Let p be t he largest divisor of the order of G, ⑴for any non-nilpotent maximal subgr oup M of G in F G*={M｜M contains a normalizer of a Sylow- p subgroup of G, and M is c-maximal}, if G satisfyi ng one of the following three conditions: (a) η(G∶M)=｜G∶M｜;(b) η(G∶M) is square-free;(c)η(G∶M) is a power of a prime;then G is p-solvable. ⑵ the following are equivalent:①G is solva ble;②η(G∶M)=｜G∶M｜ for any non-nilpotent maximal subgroup M o f G in F′ G∩F p; ③ η(G∶M) is a power of a prime for any non-nilpotent maximal subgroup M of G in F′ G∩F p.