On finite groups in which all minimal subgroups are BNA-subgroups

Abstract

A subgroup $H$ of a group $G$ is said to be a BNA-subgroup of $G$ if either $H^x=H$ or $x\in ⟨H,H^x⟩$ for all $x\in G$. The purpose of this paper is first to give the best bound for the Fitting height of $G$ if all minimal subgroups of $G$ are BNA-subgroups of $G$, and next to give an answer to the question of He, Li, and Wang [Rend. Semin. Mat. Univ. Padova 136 (2016), 51–60]. Finally, we use a few BNA-subgroups of prime order to determine the structure of the finite groups. In fact, some new conditions for a finite group to be supersolvable have been given.