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 of Abelian groups is said to be _CH-definable in the class if, for any group Bin, it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _CH-definable in , then the class is called an _CH-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _CH-class, where C is a completely decomposable torsion-free Abelian group.