Weakly characteristic subgroups of finite groups

Suppose that [Formula: see text] is a finite group and [Formula: see text]. We say that [Formula: see text] is called weakly characteristic in [Formula: see text] with respect to [Formula: see text] if for all subgroups [Formula: see text] of [Formula: see text] containing [Formula: see text], we have [Formula: see text] char [Formula: see text]. In particular, when [Formula: see text] is a subgroup of prime power order and [Formula: see text] is a Sylow subgroup containing it, [Formula: see text] is simply said to be a weakly characteristic subgroup of [Formula: see text] or weakly characteristic in [Formula: see text]. In this paper, we investigate the structure of finite groups by means of weakly characteristic subgroups.