Finite groups with Frobenius condition for non-normal primary subgroups

Abstract

A finite group P is said to be primary if for some prime p. We say a primary subgroup P of a finite group G satisfies the Frobenius condition in G if is a p-group provided P is p-group. In this article, we determine the structure of a finite group G in which every non-subnormal primary subgroup satisfies the Frobenius condition. In particular, we prove that if every non-normal primary subgroup of G satisfies the Frobenius condition, then is abelian and every maximal non-normal nilpotent subgroup U of G with is a Carter subgroup of G.