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 ∈ ⟨ 𝐻, 𝐻 𝑥 ⟩ for all 𝑥 ∈ 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 give an answer for the question of the paper [On 𝐵𝑁𝐴 -normality and solvability of finite groups, Rend. Sem. Mat. Univ. Padova 136 (2016), 51-60]. Finally we use 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.