An exact upper bound for the sum of powers of element orders in non-cyclic finite groups

Abstract

For a finite group G, let ψ(G) denote the sum of element orders of G. This function was introduced by H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs in 2009 and they proved that for any finite group G of order n, ψ(G) is maximum if and only if G≃Zn where Zn denotes the cyclic group of order n. Furthermore, M. Herzog, P. Longobardi, and M. Maj in 2018 proved that if G is non-cyclic, ψ(G)≤711ψ(Zn). S. M. Jafarian Amiri and M. Amiri in 2014 introduced the function ψk(G) which is defined as the sum of the k-th powers of element orders of G and they showed that for every positive integer k, ψk(G) is also maximum if and only if G is cyclic.In this paper, we have been able to prove that if G is a non-cyclic group of order n, then ψk(G)≤1+3.2k1+2.4k+2kψk(Zn). Setting k=1 in our result, we immediately get the result of Herzog et al. as a simple corollary. Besides, a recursive formula for ψk(G) is also obtained for finite abelian p-groups G, using which one can explicitly find out the exact value of ψk(G) for finite abelian groups G.