The influence of SS-quasinormality of some subgroups on the structure of finite groups.
The authors take up the concept of S-quasinormality occurring in \textit{O. H. Kegel} [Math. Z. 78, 205-221 (1962; Zbl 0102.26802)] and now generalise it to SS-quasinormality: a subgroup \(H\) of a finite group \(G\) is defined as being an SS-quasinormal subgroup of \(G\) if \(H\) is supplemented by, \(B\) say, in \(G\) such that \(H\) permutes with every Sylow subgroup of \(B\) (Definition 1.1). That new concept is being used to derive a number of sufficient conditions for groups being \(p\)-nilpotent (Theorem 1.1, 1.2, 1.7), \(p\)-supersoluble (Theorem 1.3, 1.4), belonging to saturated formations containing the class of supersoluble groups (Theorem 1.5), and belonging to classes of groups with the Sylow tower property of supersoluble type (Theorem 1.8). Throughout the paper, SS-quasinormality is used together with the concept of \(\mathcal M(G)\), the family of all maximal subgroups of all Sylow subgroups of \(G\); moreover, if \(P\) is a \(p\)-group and \(d\) the minimal number of generators of \(P\), that of \(\mathcal M_d(P)=\{P_1,\dots,P_d\}\), a set of maximal subgroups of \(P\) such that \(\bigcap^d_{i=1}P_i\) equals the Frattini subgroup of \(P\).
scientific article; zbMATH DE number 5304648
scientific article