Fuchs’ problem 34 for mixed Abelian groups
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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.References
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Additional Information
- Ulrich Albrecht
- Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
- Email: albreuf@auburn.edu
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1021-1029
- MSC (1991): Primary 20K15, 20K30; Secondary 20J05
- DOI: https://doi.org/10.1090/S0002-9939-02-06612-1
- MathSciNet review: 1948091