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This paper investigates self-small abelian groups of finite torsion-free rank. We obtain a new characterization of infinite self-small groups. In addition, self-small groups of torsion-free rank 1 and their finite direct sums are discussed.2000 Mathematics subject classification: primary 20K21.

Abstract. 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 abelian groups. In particular, we address Problem 34 in Professor Fuchs' book which asks if 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 X which determines the torsion-free groups of finite rank up to quasi-isomorphism.

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