Some properties on sums of element orders in finite p-groups

In literature, there are many papers on the sum of element orders of a finite group. In this study, in particular, we deal with the cases 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 such sums. 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 .