Finite groups with some \(\mathcal H\)-subgroups. (Q716494)

Let \(G\) be a finite group. A subgroup \(H\) of \(G\) is called an \(\mathcal H\)-subgroup if \(N_G(H)\cap H^g\leq H\) for all \(g\) in \(G\). This subgroup embedding property was introduced by \textit{M. Bianchi, A. Gillio Berta Mauri, M. Herzog} and \textit{L. Verardi} [J. Group Theory 3, No. 2, 147-156 (2000; Zbl 0959.20024)] to characterize finite soluble groups in which normality is a transitive relation. The main results of the paper under review solve some open questions related to this subgroup embedding property, showing that if the family of the subgroups of the same order of the Sylow subgroups are \(\mathcal H\)-subgroups, then the structure of the group is quite restricted. Theorem~3.2 is typical: Let \(\mathfrak F\) be a saturated formation containing the class of all supersoluble groups, and let \(E\) be a normal subgroup of a group \(G\) containing the \(\mathfrak F\)-residual of \(G\). Then the group \(G\) belongs to \(\mathfrak F\) if every non-cyclic Sylow subgroup of \(E\) has a family of subgroups of the same order which are \(\mathcal H\)-subgroups of \(G\).