On groups with average element orders equal to the average order of the alternating group of degree \(5\)

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 average element orders, i.e. the quotient \(o(G)=\frac{\psi(G)}{|G|}.\) We prove that \(o(G) = o(A_5)\) if and only if \(G \simeq A_5\), where \(A_5\) is the alternating group of degree \(5\).