On ℋN-subgroups of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we introduce a new subgroup embedding property generalizing the concept of [Formula: see text]-subgroups. Namely, we define a subgroup [Formula: see text] of a finite group [Formula: see text] to be an [Formula: see text]-subgroup of [Formula: see text] if there is a normal subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]. We use this concept to prove some new criteria for [Formula: see text]-nilpotence and supersolvability of finite groups. Several previously known results are generalized and extended by our results.