Fuchs’ problem 34 for mixed Abelian groups

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.