Abstract
Let H be a subgroup of a group G. We say that H satisfies Π-property in G if |G/K: NG/K(HK/K ∩ L/K)| is a π(HK/K ∩ L/K)-number for any chief factor L/K of G. If there is a subnormal supplement T of H in G such that H ∩ T ≤ I ≤ H for some subgroup I satisfying Π-property in G, then H is said to be Π-normal in G. Using these properties that hold for some subgroups, we derive new p-nilpotency criteria for finite groups.