Another criterion for solvability of finite groups

Abstract

Let G be a finite group. Denote by ψ(G) the sum ψ(G)=\sum_{x\in G}|x| where |x| denotes the order of the element x, and by o(G) the quotient o(G)=\frac{ψ(G)}{|G|}. Confirming a conjecture posed by E.I. Khukhro, A. Moreto and M. Zarrin, we prove that if o(G)< o(A_5), then G is solvable.