On 𝒮𝒮ℋ-subgroups and the structure of finite groups

A subgroup [Formula: see text] of a group [Formula: see text] is called an [Formula: see text]-subgroup in [Formula: see text] if there exists an [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text], where [Formula: see text] is the intersection of all [Formula: see text]-permutable subgroup of [Formula: see text] containing [Formula: see text]. In this paper, we present two new sufficient and necessary conditions on the [Formula: see text]-nilpotency of finite groups by use of a small quantity of [Formula: see text]-subgroups of Sylow [Formula: see text]-subgroups. A number of known results are improved and extended.