Bijections between compositions over finite groups

Abstract

One may generalize integer compositions by replacing positive integers with elements from an additive group, giving the broader concept of compositions over a group. In this note, we give some simple bijections between compositions over a finite group. It follows from these bijections that the number of compositions of a nonzero group element $g$ is independent of $g$. As a result, we derive a simple expression for the number of compositions of any given group element. This extends an earlier result for abelian groups which was obtained using generating functions and a multivariate multisection formula.