Sum of the powers of the orders of elements in finite abelian groups

Abstract

Let G be a finite group and let (G) denote the sum of element orders of G; in general l(G) denotes the the sum of the l-th powers of the element orders G where l is a positive integer. We further generalise this by introducing l(G) for negative integers l. Motivated by the recursive formula for (G), we consider a finite abelian group G and obtain a similar formula for l(G) and l(G) for l 2 (0, 1) \ Z and l 2 (-1, 0) \ Z respectively.