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 pnilpotency 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.
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