Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

A note on Fuchs' Problem 34

Author(s): U. F. Albrecht; H. P. Goeters
Journal: Proc. Amer. Math. Soc. 124 (1996), 1319-1328.
MSC (1991): Primary 20K15, 20K30; Secondary 20J05
MathSciNet review: 1326993
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Abstract: 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.

References:

1.: R. A. Beaumont and R. S. Pierce, Torsion-free groups of rank two, Memoirs of the AMS, Vol. 38 (1961). MR 24:A162
2.: R. A. Beaumont and R. J. Wisner, Rings with additive group which is torsion-free of rank two, Acta. Sci. Math. (Szeged) 20 (1959), 105--116. MR 21:5651
3.: T. Faticoni and H. P. Goeters, On torsion-free $\mathrm{Ext}$ , Comm. in Algebra 16(9) (1988), 1853--1876. MR 90d:20098
4.: L. Fuchs, Infinite Abelian Groups, Vol. I/II, Academic Press, New York, London (1970/73). MR 41:333; MR 50:2362
5.: T. Jech, Set Theory, Academic Press, New York, London (1978). MR 80a:03062

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