Fuchs' problem 34 for mixed Abelian groups

This paper investigates the extent to which an Abelian group A is determined by the homomorphism groups Hom(A. G). A class C of Abelian groups is a Fuchs 34 class if A and C in C are isomorphic if and only if Hom(A,G)≅ Hom(C,G) for all G ∈ C. Two p-groups A and C satisfy Hom(A, G) ≅ Hoot(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 G introduced by Glaz and Wickless. While G is a Fuchs 34 class, the class of (adjusted) cotorsion groups is not.