On Vanishing of the Group $\mathrm{Hom}(-, C)$

It is well known that the set of homomorphisms from a fixed abelian group $A$ to a fixed abelian group $B$ forms an additive abelian group denoted as $\mathrm{Hom}(A,\,B).$ Homomorphism groups of abelian groups possess many remarkable properties. For example, they behave like functors in the category of abelian groups. In some important cases, one can express invariants of the group $\mathrm{Hom}(A,\,B)$ in terms of invariants of the groups $A$ and $B,$ e.g., if $A$ is a torsion abelian group or if $B$ is an algebraically compact abelian group. If $A=B,$ the group $\mathrm{Hom}(A,\,B)=\mathrm{End}(A,\,B)$ is called the endomorphism group of the group $A;$ it can be turned into a ring denoted as $\mathrm{E}(A).$ Studying homomorphism groups and endomorphism rings is an important problem of the theory of abelian groups. In particular, describing abelian groups such that $\mathrm{Hom}(A,\,B)=0$ is one of open problems in this theory. For example, the group $\mathrm{Hom}(A,\,B)$ is zero in the following case. Let an abelian group $G$ be decomposed into a sum of its subgroups $A$ and $B,$ $A$ being a fully invariant subgroup in the group $G,$ i.e., $A$ is mapped into itself under any endomorphism of the group $G.$ Then, $\mathrm{Hom}(A,\,B)=0.$ The torsion subgroup of a group, for example, is its fully invariant subgroup. In this paper, a criterion of vanishing is presented for an arbitrary homomorphism from an arbitrary abelian group to an arbitrary torsion free group.