A NOTE ON FUCHS PROBLEM-34

We investigate to what extent an abelian group G is determined by the homomorphism groups Hom(G, B) where B is chosen from a set X of abelia n groups. In particular, we address Problem 34 in Professor Fuchs' boo k which asks if X can be chosen in such a way that the homomorphism gr oups determine G up to isomorphism. We show that there is a negative a nswer to this question. On the other hand, there is a set X which dete rmines the torsion-free groups of finite rank up to quasi-isomorphism.