Weakly normal subgroups and classes of finite groups. (Q2856420)

The present paper deals with finite groups. The notions of ``weakly normal subgroup'' and of ``PST-group'' were historically introduced, in order to improve some classical theorems of Kegel and Wielandt on the product of finite groups. These two notions are widely illustrated in Zbl 1206.20019 [\textit{A. Ballester-Bolinches, R. Esteban-Romero, M. Mohamed}, Products of finite groups. de Gruyter Expositions in Mathematics 53. Berlin: Walter de Gruyter (2010)], and, by the way, in some fundamental contributions of the author of the paper under review.NEWLINENEWLINE An interesting feature of the topic deals with some conditions of splitting in semidirect products. Roughly speaking, if we know enough on the number of weakly normal subgroups, on their position in the subgroup lattice of a group, and, analogously, enough on the number and on the position of those subgroups which are PST-groups, then we may control the structure of the whole group in a very detailed way. This is the spirit of the paper.NEWLINENEWLINE More technically, here the author focuses on the role of the set \(S_p(G)\) of all cyclic subgroups of a group \(G\) of prime order \(p\) or of order 4; on the set \(\overline{S_p}(G)\) of all subgroups of \(G\) of prime power order; finally, on the set \(WN(G)\) of all weakly normal subgroups of \(G\). For instance, Theorem A shows that the inclusion \(S_p(G)\subseteq WN(G)\) implies that \(G\) is supersolvable. The same happens (see again Theorem A) when \(\overline{S}_p(G')\subseteq WN(G)\). Theorem B is still more specific: it describes conditions for which the previous inclusion may happen. This is equivalent to the presence of a so-called ``nilpotent Hall complement'' which satisfies certain precise properties. In particular, it is shown that we need to have a semidirect product. The remaining results, that is, Theorems C, D, E and F describe further circumstances, under the perspective of the notion of PST-group.