On Π-property of subgroups of a finite group.

Let G be a finite group, p a prime, P a Sylow p -subgroup of G and d a power of p such that 1 < d < | P |. Let G p ∗ denote the unique smallest normal subgroup of G for which the corresponding factor group is abelian of exponent dividing p − 1. Let 1 , 2 , 3 be classes of all p -groups, p -nilpotent groups and p -supersolvable groups, respectively, G be the -residual of G. Let X ∈ { (G p ∗) 1 , G 2 , G 3 }. A subgroup H of a finite group G is said to have Π -property in G , if for any G -chief factor L / K , | G / K : N G / K ((H ∩ L) K / K) | is a π ((H ∩ L) K / K) -number. A normal subgroup E of G is said to be p -hypercyclically embedded in G if every p - G -chief factor of E is cyclic, where p is a fixed prime. In this paper, we prove that E is p -hypercyclically embedded in G if and only if for some p -subgroups H of E , H ∩ X have Π -property in G. [ABSTRACT FROM AUTHOR]