p-nilpotence of finite groups and minimal subgroups

In this paper, it is proved that a finite group G is p-nilpotent if every minimal subgroup of P∩Op(G) is permutable in P and NG(P) is p-nilpotent, and when p=2 either [Ω2(P∩Op(G)),P] Ω1(P∩Op(G)) or P is quaternion-free, where p is a prime dividing the order of G and P is a Sylow p-subgroup of G. By using this result, we may get a series of corollaries for p-nilpotence, which contain some known results. Some other applications of this result are also given.