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.