Some Properties on Sums of Element Orders in Finite p-groups

Abstract

In literature, there are many papers on the sum of element orders of a finite group. In this study, in particular, we deal with thecases in finite p-groups. Our main aim is to investigate the sums of element orders in finite p-groups and to give some properties of suchsums. Let (G) denote the sum of element orders of a finite group G. As an immediate consequence, we proved that \psi(G) ≤ 3 /4 \psi(C) and\psi(G) < 1/p-1\psi (C), where G is a non-cyclic finite p-group of order p^r and C is a cyclic group of order p^r for some prime p.