Finite groups with some subgroups of prime power order satisfying the partial Π-property

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π-property in G if there exists a G-chief series G: 1 =G₀ < G₁ < ··· < Gₙ= G of G such that | G / Gᵢ₋₁ : N_G/Gᵢ₋₁ (HGᵢ₋₁/Gᵢ₋₁∩ Gᵢ/Gᵢ₋₁)| is a π (HGᵢ₋₁/Gᵢ₋₁∩ Gᵢ/Gᵢ₋₁)-number for every G-chief factor Gᵢ/Gᵢ₋₁ of G, 1≤ i≤ n. In this paper, we investigate the structure of a finite group G under the assumption that some subgroups of prime power order satisfy the partial Π-property.