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

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