Finite groups with n-embedded subgroups

Let [Formula: see text] be a finite group. How minimal subgroups can be embedded in [Formula: see text] is a question of particular interest in studying the structure of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for all Sylow subgroups [Formula: see text] of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is called [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] with [Formula: see text]-embedded subgroups.