Description of the group Hom$(A,G)$ of homomorphisms from an abelian group $A$ to an abelian group $G$?

Question

Is there a nice description of the group Hom$(A,G)$ of homomorphisms from an abelian group $A$ to an abelian group $G$? The group operation is addition of homomorphisms.

For example, one can show that Hom$(\mathbb{Z},\mathbb{Z})\approx \mathbb{Z}$.

Answer 1

If $A$ and $G$ are finitely generated, then yes. This follows from the fact that you can take finite direct sums out of either variable in $\textrm{Hom}(-,-)$, and the isomorphism

$$\textrm{Hom}(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}/m\mathbb{Z}) \cong \mathbb{Z}/(n,m)\mathbb{Z}$$

(even when $n$ or $m$ is $1$).