Sum of the Products of the Orders of Two Distinct Elements in Finite Groups

Let G be a finite group. Then we denote ψ(G) the sum of element orders in G. In 1 Amiri , H. , Jafarian Amiri , S. M. , Isaacs , I. M. ( 2009 ). Sums of element orders in finite groups . Comm. Algebra 37 ( 9 ): 2978 – 2980 .Taylor &; Francis Online], Web of Science ®] , Google Scholar]] it is proved that if G is a non-cyclic group of order n, then ψ(G) < ψ(C _n ), where C _n is the cyclic group of order n. Here we generalize and improve this result, and also we give an application of this improvement.