Geodesic
Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms
Matematičeskie zametki, Tome 73 (2003) no. 5, pp. 643-648.

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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. 
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T. A. Beregovaya; A. M. Sebel'din. Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms. Matematičeskie zametki, Tome 73 (2003) no. 5, pp. 643-648. http://geodesic.mathdoc.fr/item/MZM_2003_73_5_a0/

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