On finite groups with some Hall normally embedded subgroups

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 [Formula: see text], where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text], that is, the smallest normal subgroup of [Formula: see text] containing [Formula: see text]. A group [Formula: see text] is called a [Formula: see text]-group if its all minimal subgroups and cyclic subgroup of order [Formula: see text] are Hall normally embedded in [Formula: see text]. In this paper, we give the classification of minimal non-[Formula: see text]-groups. Furthermore, we investigate the structure of finite group all of whose maximal subgroups of even order are [Formula: see text]-groups.