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

Abstract

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