A note on Fuchs’ Problem 34

Albrecht, U.; Goeters, H.

doi:10.1090/S0002-9939-96-03324-2

A note on Fuchs’ Problem 34
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by U. F. Albrecht and H. P. Goeters PDF

Proc. Amer. Math. Soc. 124 (1996), 1319-1328 Request permission

Abstract:

We investigate to what extent an abelian group $G$ is determined by the homomorphism groups $\operatorname {Hom}(G,B)$ 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.

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