Finite groups with specific \(S_*\)-embedded subgroups (Q6635334)

Let \(G\) be a finite group. \textit{C. Li} [Commun. Algebra 50, No. 4, 1585--1594 (2022; Zbl 07488684)] introduced a new embedding property: a subgroup \(A\) of \(G\) is said to be \(S_{\ast}\)-embedded in \(G\) if \(G\) has an \(S\)-permutable subgroup \(F\) such that \(AF\) is \(S\)-permutable in \(G\) and \(A \cap F \leq A_{\ast}\), where \(A_{\ast}\) is a subgroup contained in \(A\) which is either \(S\)-permutably embedded or \(S\)-semipermutable in \(G\).\N\NIn the paper under review, the authors study the \(p\)-nilpotency and supersolvablity of a finite group \(G\) under the assumption that certain subgroups of prime power order are \(S_{\ast}\)-embedded in \(G\). They extend and generalize some results of Li [loc. cit.].