On weakly ℋC-embedded subgroups and p-nilpotence of finite groups

Abstract

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]; [Formula: see text] is called weakly [Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text]. In this paper, we study the [Formula: see text]-nilpotence of a group [Formula: see text] in which every subgroup of order [Formula: see text] of a Sylow [Formula: see text]-subgroup [Formula: see text] with [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Many recent results in the literature related to [Formula: see text]-nilpotence of [Formula: see text] are generalized.