The influence of weakly \(s\)-permutably embedded subgroups on the \(p\)-nilpotency of finite groups

Abstract

Suppose \(G\) is a finite group and \(H\) is a subgroup of \(G\). \(H\) is said to be \(s\)-permutably embedded in \(G\) if for each prime \(p\) dividing \(|H|\), a Sylow \(p\)-subgroup of \(H\) is also a Sylow \(p\)-subgroup of some \(s\)-permutable subgroup of \(G\); \(H\) is called  weakly \(s\)-permutably embedded in \(G\) if there are a subnormal subgroup  \(T\) of \(G\) and an \(s\)-permutably embedded subgroup \(H_{se}\) of \(G\) contained in \(H\) such that \(G=HT\) and \(H\cap T\leq H_{se}\). We investigate the influence of weakly \(s\)-permutably embedded subgroups on the \(p\)-nilpotency of finite groups.