A note on Fuchs’ Problem 34

Abstract

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