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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on Fuchs' Problem 34


Authors: U. F. Albrecht and 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.


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Additional Information

U. F. Albrecht
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: albreuf@mail.auburn.edu

H. P. Goeters
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: goetehp@mail.auburn.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03324-2
PII: S 0002-9939(96)03324-2
Keywords: Homomorphism group, Martin's Axiom, $p$-group, torsion-free group
Received by editor(s): October 1, 1993
Received by editor(s) in revised form: December 12, 1993
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1996 American Mathematical Society