On \(\pi\)-quasinormally embedded subgroups of finite group.
All groups considered in this review are finite. The study of the influence of the embedding properties of subgroups on the structure of the groups plays an important role in clarifying the structure of the composite groups. The paper under review is a nice contribution in this line. The authors study the influence of the \(\pi\)-quasinormal embedding of some relevant families of subgroups on the structure of the group within the framework of formation theory. This subgroup embedding property was introduced by \textit{M. C. Pedraza-Aguilera} and the reviewer [J. Pure Appl. Algebra 127, No. 2, 113-118 (1998; Zbl 0928.20020)]. The following result is typical: Let \(\mathcal F\) be a saturated formation containing all the supersoluble groups. Then a group \(G\) belongs to \(\mathcal F\) if and only if there is a normal subgroup \(H\) in \(G\) such that \(G/H\) belongs to \(\mathcal F\) and the subgroups of prime order or order \(4\) (respectively, the maximal subgroups of Sylow subgroups) of the generalised Fitting subgroup of \(H\) are \(\pi\)-quasinormally embedded in \(G\). The results of the paper extend and improve earlier results of Asaad and Heliel and also more recent results of the authors.
scientific article; zbMATH DE number 2134418
scientific article