On the Question of the Definability of Certain Classes of Completely Decomposable Abelian Torsion-Free Groups by Their Homomorphism Groups

Let $C$ be an Abelian group. A class $X$ is said to be a $_{C}H$-class if, for any groups $A,B\in X$, a group isomorphism of $\operatorname{Hom}(C,A)$ and $\operatorname{Hom}(C,B)$ implies an isomorphism of the groups $A$ and $B$. In the paper, conditions on a completely decomposable Abelian group $C$ are investigated under which a class of certain completely decomposable torsion-free Abelian groups is a $_{C}H$-class.