Finite group with given Hall normally embedded subgroups

Abstract

Let G be a finite group. A subgroup H of G is called Hall normally embedded in G if H is a Hall subgroup of the normal closure \(H^G\). In this paper, we fix a subgroup D of Sylow p-subgroup P of G with \(1<|D|<|O_p(G)|\) and study the structure of G under the assumption that all subgroups H of P with order \(|H|=|D|\) are Hall normally embedded in G.