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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.
Yu et al. (Wed,) studied this question.