  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 $\Gamma$.
