Finite groups with Frobenius condition for non-normal primary subgroups

摘要

A finite groupPis said to beprimaryif|P|=pafor some primep. We say a primary subgroupPof a finite groupGsatisfies theFrobenius conditioninGifNG(P)/CG(P)is ap-group providedPisp-group. In this article, we determine the structure of a finite groupGin which every non-subnormal primary subgroup satisfies the Frobenius condition. In particular, we prove that if every non-normal primary subgroup ofGsatisfies the Frobenius condition, thenG/F(G)is abelian and every maximal non-normal nilpotent subgroupUofGwithF(G)U=Gis a Carter subgroup ofG.