A sufficient and necessary condition for p-nilpotence of a finite group

In this paper, we introduce a condition on a [Formula: see text]-subgroup of a finite group that is more general than the concept of [Formula: see text]-subgroups, and obtain a sufficient and necessary condition for [Formula: see text]-nilpotence of a finite group. We not only simplify the proofs of some main theorems of Gao and Li [Acta Math. Hungar. 163(1) (2021) 29–36], Kaspczyk [Acta Math. Hungar. 165(2) (2021) 355–359], and Kaspczyk [J. Algebra Appl. 23(1) (2024) 2450016], but also generalize these results.