On the divisibility of $#\Hom(\Gamma,G)$ by $|G|

TL;DR

This paper investigates conditions under which the number of homomorphisms from a finitely generated group to a finite group is divisible by the order of the finite group, linking this to the group's abelianization.

Contribution

It extends Solomon's result by characterizing when the divisibility holds for all finite groups based on the group's abelianization properties.

Findings

Divisibility of homomorphism counts relates to infinite abelianization.

Characterization of groups with divisibility property for all finite groups.

Arithmetic properties of subgroup counts derived from the main result.

Abstract

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $\Gamma$ is a finitely generated group, then $|G|$ divides $#\Hom(\Gamma,G)$ for every finite group $G$ if and only if $\Gamma$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $\Gamma$.