Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let $C$ be an Abelian group. An Abelian group $A$ in some class $\mathscr X$ of Abelian groups is said to be $\sideset{_C}{}{\mathop H}$-definable in the class $\mathscr X$ if, for any group $B\in\mathscr X$, it follows from the existence of an isomorphism $\operatorname{Hom}(C,A)\cong\operatorname{Hom}(C,B)$ that there is an isomorphism $A\cong B$. If every group in $\mathscr X$ is ${}_CH$-definable in $\mathscr X$, then the class $\mathscr X$ is called an ${}_CH$-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a $\sideset{_C}{}{\mathop H}$-class, where $C$ is a completely decomposable torsion-free Abelian group.