z-permutable subgroups and p-nilpotency 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 characterize p-nilpotency of finite groups G with assumption that some maximal subgroups or some 2-maximal subgroups of Sylow subgroups of G are Z-permutable. Some recent results are extended.