On \(S\)-quasinormally embedded subgroups of finite groups

Abstract

Let \(G\) be a finite group. A subgroup \(A\) is called: 1) \(S\)-quasinormal in \(G\) if \(A\)  is permutable with all Sylow subgroups in  \(G\) 2) \(S\)-quasinormally embedded in \(G\) if every Sylow subgroup of \(A\) is a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). Let \(B_{seG}\) be the subgroup  generated by all the subgroups of \(B\)  which are \(S\)-quasinormally embedded in \(G\). A subgroup \(B\) is called  \(SE\)-supplemented in  \(G\) if there exists a subgroup \(T\) such that \(G=BT\) and \(B\cap T\le B_{seG}\).  The main result of the paper is the following.

Theorem. Let \(H\) be a normal subgroup in \(G\), and \(p\) a prime divisor of \(|H|\) such that  \((p-1,|H|)=1\). Let \(P\) be a Sylow \(p\)-subgroup in \(H\). Assume that  all maximal subgroups in \(P\)   are \(SE\)-supplemented in \(G\). Then \(H\) is  \(p\)-nilpotent and all its \(G\)-chief \(p\)-factors are cyclic.