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Fuchs' problem 34 for mixed Abelian groups

Authors:	Ulrich Albrecht

Institution:	Department of Mathematics, Auburn University, Auburn, Alabama 36849

Abstract:	This paper investigates the extent to which an Abelian group is determined by the homomorphism groups $\operatorname{Hom}(A,G)$ . A class $\mathcal C$ of Abelian groups is a Fuchs 34 class if and 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 -groups and satisfy $\operatorname{Hom}(A,G) \cong \operatorname{Hom}(C,G)$ for all groups 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.

Keywords:	Homomorphism group $p$-group mixed group

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