On weakly \(s\)-quasinormally embedded and \(ss\)-quasinormal subgroups of finite groups. (Q638749)

Many papers have dealt with the influence of subgroup embedding properties on the structure of a finite group. Recall that a subgroup \(H\) of a finite group \(G\) is said to be \(s\)-quasinormally embedded (or \(s\)-permutably embedded) in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of an \(s\)-quasinormal (or \(s\)-permutable) subgroup of \(G\) [see \textit{A. Ballester-Bolinches} and \textit{M. C. Pedraza-Aguilera}, J. Pure Appl. Algebra 127, No. 2, 113-118 (1998; Zbl 0928.20020)]. On the other hand, \textit{Y. Li, S. Qiao} and \textit{Y. Wang} [Commun. Algebra 37, No. 3, 1086-1097 (2009; Zbl 1177.20036)] took up this concept and generalised it to `weakly \(s\)-quasinormal embedding': a subgroup \(H\) of a finite group \(G\) is said to be weakly \(s\)-quasinormally embedded in \(G\) if there exists a subnormal subgroup \(T\) of \(G\) and an \(s\)-permutably embedded subgroup \(H_{se}\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{se}\). A subgroup \(H\) of \(G\) is said to be \(ss\)-quasinormal if there is a subgroup \(B\) of \(G\) such that \(G=HB\) and \(H\) permutes with every Sylow subgroup of \(B\) [see \textit{S. Li, Z. Shen, J. Liu} and \textit{X. Liu}, J. Algebra 319, No. 10, 4275-4287 (2008; Zbl 1152.20019)]. The above two concepts are being used to derive a number of sufficient conditions for groups being \(p\)-nilpotent (Theorem 3.1, 3.7), supersoluble (Theorem 4.12), and belonging to saturated formations containing the class of supersoluble groups (Theorem 4.1, Theorem 4.13).