On minimal subgroups of finite groups (Q2765939)

An interesting line of research in finite groups is the study of the influence the minimal subgroups (subgroups of prime order) have on the structure of the group. A classical result in this context is a theorem of Itô [see \textit{B. Huppert}, Endliche Gruppen I (1968; Zbl 0217.07201)] stating that a group \(G\) of odd order is nilpotent provided that every minimal subgroup is central in \(G\).NEWLINENEWLINENEWLINEA natural question in this context is the following: which are the embedding properties of the minimal subgroups of a group to ensure that the group belongs to a certain formation? \textit{J. Buckley} [Math. Z. 116, 15-17 (1970; Zbl 0202.02303)], \textit{M. Asaad, A. Ballester-Bolinches} and \textit{M. C. Pedraza-Aguilera} [Commun. Algebra 24, No. 8, 2771-2776 (1996; Zbl 0856.20015)] have made contributions to the above problem considering normality and Sylow-permutability.NEWLINENEWLINENEWLINEIn the paper under review the authors use the c-normality introduced by \textit{Y. Wang} [J. Algebra 180, No. 3, 954-965 (1996; Zbl 0847.20010)] to obtain new results. The authors prove among other theorems that if \(\mathcal F\) is a saturated formation containing the supersoluble groups and \(G\) is a group with a normal subgroup \(N\) such that \(G/N\) belongs to \(\mathcal F\), then \(G\) belongs to \(\mathcal F\) provided that every minimal subgroup and every subgroup of order \(4\) of \(N\) is c-normal in \(G\).