关于s-正规子群与正规指数

On s-normal Subgroups and Normal Index

利用s-正规子群与正规指数的概念给出群为可解群的一些条件.主要定理有:(1)设M是群G的可解的极大子群,M在G中s-正规的充要条件是η(G∶M)=|G∶M|;(2)有限群G可解当且仅当对于M∈F1(G)={M<.G|η(G∶M)≠|G∶M|},M在G中s-正规.(3)设N是G的正规子群,N可解的充要条件是对于任意不包含N的c-极大子群M,有η(G∶M)=|G∶M|.

In this paper,we use s-normal subgroups and normal index to discuss the solvability of a finite group and obtain some new conditions.Main results are（1）M is a solvable maximal subgroup of G,M is s-normal in G if and only if η（G∶ M）=｜G∶ M｜;（2）let G be a finite group,denote F1（G）={M·G｜η（G∶ M）≠｜G∶ M｜},G is solvable if and only if M belong to F1（G） and M is s-normal in G;（3）let G be a finite group,N is a normal subgroup of G.Then N is solvable if and only if every c-maximal subgroups of G not containing N is η（G∶ M）=｜G∶ M｜ in G.