Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.
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.
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 the normal closure [Formula: see text]. In this paper, we investigate the structure of a finite group [Formula: see text] under the assumption that certain subgroups of prime power order are Hall normally embedded in [Formula: see text].
Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be a BNA-subgroup of [Formula: see text] if either [Formula: see text] or [Formula: see text] for all [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be a weakly BNA-subgroup of [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] is a BNA-subgroup of [Formula: see text]. In this paper, we investigate the structure of a finite group [Formula: see text] under the assumption that every minimal subgroup of [Formula: see text] not having a supersolvable supplement in [Formula: see text] is a weakly BNA-subgroup of [Formula: see text].
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