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

Let (G) be a finite group. Denote by (psi(G)) the sum (psi(G)=sum_{xin 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)