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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

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    It would be sufficient if, for each fixed $n$, the subgroup of $D$ consisting of elements of order dividing $n$ was finite. That is satisfied by ${\mathbb C}^\times$.2012-12-24

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For an integer $m$, let $D_m$ be the set of $x\in D$ such that $mx=0$. Let $P(G)$ be the set of primes $p$ such that $\mathrm{Hom}(G,\mathbf{Z}/p\mathbf{Z})$ is nonzero. Note that $P(G)$ is finite since it consists of some of the prime divisors of $|G|$. Then $\mathrm{Hom}(G,D)$ is finite if and only if for every $p\in P(G)$, the subgroup $D_p$ is finite.

This characterization is true for an arbitrary abelian group $D$, by the way.

Indeed if $D_p$ is finite and let $n$ be maximal such that $G$ has an element of order $p^n$. Then for every homomorphism of $G$ into $D$, this maps $G$ into the torsion of $D$ and the projection on the $p$-part maps into $D_{p^n}$. A simple argument (for an arbitrary abelian group) shows that $D_p$ finite implies $D_{p^n}$ finite. So, writing $n=n(p)$, every homomorphism $G\to D$ maps into $\sum_{p\in P(G)}D_{p^{n(p)}}$, which is finite.

The converse (if $p\in P(G)$ and $D_p$ is infinite then $\mathrm{Hom}(G,D)$ is infinite) is immediate.