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

 

Fuchs' problem 34 for mixed Abelian groups


Author: Ulrich Albrecht
Journal: Proc. Amer. Math. Soc. 131 (2003), 1021-1029
MSC (1991): Primary 20K15, 20K30; Secondary 20J05
Published electronically: August 19, 2002
MathSciNet review: 1948091
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Abstract: This paper investigates the extent to which an Abelian group $A$ is determined by the homomorphism groups $\operatorname{Hom}(A,G)$. A class $\mathcal C$ of Abelian groups is a Fuchs 34 class if $A$ and $C$ in $\mathcal C$ are isomorphic if and only if $\operatorname{Hom}(A,G) \cong \operatorname{Hom}(C,G)$ for all $G \in \mathcal C$. Two $p$-groups $A$ and $C$ satisfy $\operatorname{Hom}(A,G) \cong \operatorname{Hom}(C,G)$ for all groups $G$ if and only if they have the same $n^{th}$-Ulm-Kaplansky-invariants and the same final rank. The mixed groups considered in this context are the adjusted cotorsion groups and the class $\mathcal G$ introduced by Glaz and Wickless. While $\mathcal G$ is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.


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Additional Information

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

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06612-1
PII: S 0002-9939(02)06612-1
Keywords: Homomorphism group, $p$-group, mixed group
Received by editor(s): June 26, 2001
Received by editor(s) in revised form: October 30, 2001
Published electronically: August 19, 2002
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2002 American Mathematical Society