某些有限群的非单性

Non-Simplicity of Certain Finite Groups

本文证明了下述定理:定理令 G 为有限群,K 和 L 是 G 的两个极大子群。如果 G 的每个真局部子群共轨地包含在 K 或 L 中,那么 G 的 Fitting 子群 F(G)≠1。特别地,G 不是非交换单群。这个定理推广了G.Pazderski 的结果:至多含有两个极大子群共轭类的有限群可解。

In this paper the following theorem is proved:THEOREM Let G be a finite group.Suppose that G has two maximum subgroups K and L.If every proper local subgroup of G is contained in K^x or L^x for some x ∈ G,Then the Fitting subgroup F(G)≠1.In particular,G is not simple. This theorem is a generalization of G.Pazderski’s result,which states that a finite group is solvable if it has at most two conjugacy classes of maximum subgroups.