A note on σ-nilpotency of finite groups

Let σ=\σᵢ i∈ I\ is some partition of the set P of all primes, and σ(n) =\σᵢ i∈ I, σᵢ∩π(n)≠∅\ for any integer n. A group G is called σ-primary if either $G =1$ or |σ(G)| =1. G is σ-nilpotent if (H/K) (G/CG(H/K)) is σ-primary for every chief factor $H/K$ of G.In this paper, we prove that G is σ-nilpotent if and only if G is a σ-full group and π(|xy|)=π(|x||y|) for any two elements x,y∈ G such that σ(|x|)∩σ(|y|)=∅. MSC(2020): 20D10, 20D20