On BNA-normality and solvability of finite groups

Abstract

Let $G$ be a finite group. A subgroup $H$ of $G$ is called a BNA-subgroup if either $H^x=H$ or $x\in ⟨H,H^x⟩$ for all $x\in G$. In this paper, some interesting properties of BNA-subgroups are given and, as applications, the structure of the finite groups in which all minimal subgroups are BNA-subgroups have been characterized.