A note on Fuchs' Problem 34

We investigate to what extent an abelian group $G$ is determined by the homomorphism groups $\mathrm{Hom}(G,B)$ where $B$ is chosen from a set $\mathcal X$ of abelian groups. In particular, we address Problem 34 in Professor Fuchs' book which asks if $\mathcal X$ can be chosen in such a way that the homomorphism groups determine $G$ up to isomorphism. We show that there is a negative answer to this question. On the other hand, there is a set $\mathcal X$ which determines the torsion-free groups of finite rank up to quasi-isomorphism.