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

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