A note on Hall normally embedded subgroups of finite groups

Abstract

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called Hall normally embedded in [Formula: see text] if [Formula: see text] is a Hall subgroup of the normal closure [Formula: see text]. In this paper, we fix a subgroup [Formula: see text] of Sylow subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that all subgroups [Formula: see text] of [Formula: see text] with [Formula: see text] are Hall normally embedded in [Formula: see text].