Finite groups with minimal weakly BNA-subgroups (Q6601857)

All groups appearing in this review are finite.\N\NMany authors have examined the structure of a finite group \(G\) under the assumption that certain subgroups of \(G\) of prime power order are well situated in \(G\).\N\N\textit{X. He} et al. [Rend. Semin. Mat. Univ. Padova, 136, 51--60 (2016; Zbl 1368.20010)] introduced the following definition: 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 \(x \in G\), \(H\) is also said to be BNA-normal in \(G\).\N\NRecently, \textit{Q. Guo} et al. [Commun. Algebra, 55, No. 11, 4746--4755 (2022; Zbl 1514.20064)] introduced the following generalization of BNA-sub\-group (they investigate the solvability, supersolvability and \(p\)-nilpotency of a finite group under the assumption that certain minimal subgroups are weakly BNA-subgroups of \(G\)): A subgroup \(H\) of a group \(G\) is said to be a weakly BNA-subgroup of \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(G=HT\) and \(H \cap T \in\mathcal{B}(G)\), the set of all BNA-subgroups of \(G\), \(H\) is also called to be weakly BNA-normal in \(G\).\N\NThe aim of the paper under review is to generalize the above mentioned results. Every \(c\)-normal subgroup and every BNA-subgroup of \(G\) are weakly BNA-normal in \(G\), but the authors present examples which show that the converse is not true in general. They investigate the structure of a finite group with the assumption that some cyclic subgroups of \(G\) of prime order or order \(4\) not having a supersolvable supplement are weakly BNA-normal in \(G\) and extend the results to a saturated formation. Their results improve and extend some recent results concerning \(c\)-normal and BNA-subgroups.