Sum of the Element Orders in Groups of the Square-Free Orders

Given a finite group G, we denote by Psi( G) the sum of the element orders in G. In this article, we prove that if t is the number of nonidentity conjugacy classes in G, then Psi( G) = 1 + t| G| if and only if G is either a group of prime order or a nonabelian group of the square-free order with two prime divisors. Also we find a unique group with the second maximum sum of the element orders among all finite groups of the same square-free order.