A criterion of p-hypercyclically embedded subgroups of finite groups

Let G be a finite group, E a normal subgroup of G and p a fixed prime. We say that E is p-hypercyclically embedded in G if every p-G-chief factor of E is cyclic. A subgroup H of G is said to satisfy Π-property in G   if |G/K:NG/K((H∩L)K/K)||G/K:NG/K((H∩L)K/K)| is a π((H∩L)K/K)π((H∩L)K/K)-number for any chief factor L/KL/K in G; H is said to be Π-normal in G if G has a subnormal supplement T of H   such that H∩T⩽I⩽HH∩T⩽I⩽H for some subgroup I satisfying Π-property in G. In this paper, we prove that E is p-hypercyclically embedded in G if and only if some classes of p-subgroups of E are Π-normal in G. Many recent results are extended.