Finite groups with \(\mathcal H\)-subgroups or strongly closed subgroups. (Q2882899)

Let \(G\) be a finite group. \textit{M. Bianchi, A. Gillio Berta Mauri, M. Herzog} and \textit{L. Verardi} introduced [in J. Group Theory 3, No. 2, 147-156 (2000; Zbl 0959.20024)] the following notion: A subgroup \(H\) of \(G\) is called an \(\mathcal H\)-subgroup if \(N_G(H)\cap H^g\leq H\) for all \(g\) in \(G\). On the other hand, \textit{D. M. Goldschmidt} [in Ann. Math. (2) 99, 70-117 (1974; Zbl 0276.20011)] and \textit{R. J. Flores} and \textit{R. M. Foote} [in Adv. Math. 222, No. 2, 453-484 (2009; Zbl 1181.20014)] investigated the following concept: Let \(S\) be a subgroup of \(G\); a subgroup \(A\) of \(S\) is said to be strongly closed in \(S\) with respect to \(G\) if for every \(a\in A\) it holds that \(a^G\cap S\leq A\), where \(a^G\) denotes the \(G\)-conjugacy class of \(a\). In particular, when \(A\) is of prime-power order and \(S\) is a Sylow subgroup containing it, \(A\) is simply said to be a strongly closed subgroup. It turns out that \(\mathcal H\)-subgroups of prime-power order are exactly the same as strongly closed subgroups. Note also that Sylow \(p\)-subgroups are \(\mathcal H\)-subgroups.NEWLINENEWLINE In this paper, the authors study groups such that every cyclic subgroup of prime order or of order 4 is an \( \mathcal H\)-subgroup, which are called \(P\mathcal H\)-groups. More concretely, they analyze the structure of minimal non-\(P\mathcal H\)-groups (i.e. non-\(P\mathcal H\)-groups all of whose proper subgroups are \(P\mathcal H\)-groups). Among other results, they show that the structure of minimal non-\(P\mathcal H\)-groups is the same as that of minimal non-\(\mathcal T\)-groups (a \(\mathcal T\)-group is a group for which normality is a transitive relation).