Another criterion for solvability of finite groups

TL;DR

This paper proves that a finite group G is solvable if the average element order, measured by o(G), is less than that of the group A_5, confirming a conjecture in group theory.

Contribution

It confirms a conjecture linking the average element order of a finite group to its solvability, providing a new criterion for solvability.

Findings

If o(G)<o(A_5), then G is solvable.

The conjecture by Khukhro, Moreto, and Zarrin is validated.

The result offers a new solvability criterion based on element orders.

Abstract

Let G be a finite group. Denote by \psi(G) the sum \psi(G)=\sum_{x\in G}|x| where |x| denotes the order of the element x, and by o(G) the quotient o(G)=\frac{\psi(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.