On finite groups with Hall normally embedded Schmidt subgroups

Abstract

A subgroup \(H\) of a finite group \(G\) is said to be Hall normally embedded in \(G\) if there is a normal subgroup \(N\) of \(G\) such that \(H\) is a Hall subgroup of \(N\). A Schmidt group is a non-nilpotent finite group whose all proper subgroups are nilpotent. In this paper, we prove that if each Schmidt subgroup of a finite group \(G\) is Hall normally embedded in \(G\), then the derived subgroup of \(G\) is nilpotent.