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 \langle H, H^{x}\rangle 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.