Influence of Z-permutable of Maximal Subgroups of Sylow Subgroups of Finite Groups

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable in G if H permutes with every member of ℨ. In this paper, we prove the p-nilpotency of a finite group with assumption that some subgroups of Sylow subgroup are ℨ-permutable in the normalizers of Sylow subgroups. Our results unify and generalize some earlier results.