A subgroup H of a finite group G is said to be W-S-permutable in G if there is a subgroup K of G such that G = HK and H ∩ K is a nearly S-permutable subgroup of G. In this article, we analyse the structure of a finite group G by using the properties of W-S-permutable subgroups and obtain some new characterizations of finite p-nilpotent groups and finite supersolvable groups. Some known results are generalized.
A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be an [Formula: see text]-subgroup in [Formula: see text] if [Formula: see text] for all [Formula: see text]; and [Formula: see text] is said to be a weakly [Formula: see text]-subgroup in [Formula: see text] if there is a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is an [Formula: see text]-subgroup of [Formula: see text]. In this paper, we give a positive answer to a problem posed by Li and Qiao [On weakly [Formula: see text]-subgroups and [Formula: see text]-nilpotency of finite groups, J. Algebra Appl. 16 (2017) 1750042].
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