If $G$ is a finite group, and $D$ is a divisible abelian group, what are some conditions on $D$ for which $\mathrm{Hom}(G,D)$ is finite?
At first I thought that having $D$ with finite torsion subgroup was as good as it gets, but then looking around I found that for $D=\mathbb{C}^{\times}$, the Hom group is also finite but the torsion of $\mathbb{C}^{\times}$ is not finite. But for example I asked this question before for all divisible groups and I since know that if $D$ consisted of infinitely many copies of $\mathbb{Q}/ \mathbb{Z}$ then the Hom group would be infinite.
So what are some of the least restrictive conditions I can put on $D$ to make the Hom group finite?
Thank you