On the isomorphism of the group of homomorphisms of two torsion-free Abelian groups to one of these groups (Q1909804)

The authors deal with the question when \(\text{Hom}(A,B)\cong B\) for a completely decomposable torsion-free abelian group \(A\). As a sample result the last of four theorems is stated below. Theorem. Let \(A=\bigoplus_{i\in I}A_i\) be the direct sum of the finitely many rank-one groups \(A_i\), \(i\in I\). Suppose that \(B=B_1\oplus B_2\) where \(B_i\) is \(\tau_i\)-homogeneous, the types \(\tau_1\) and \(\tau_2\) are incomparable and \(B_i\oplus B_i\cong B_i\) for \(i=1,2\). Then \(\text{Hom}(A,B)\cong B\) if and only if 1. and 2. hold. 1. There are indices \(i_1\), \(i_2\) such that \(\text{type}(A_{i_1})\leq\tau_1\) and \(\text{type}(A_{i_2})\leq\tau_2\); 2. For \(k=1\) or \(k= 2\), if \(\text{type}(A_i)\leq\tau_k\), then \(\text{type}(A_i)\) contains a characteristic \(\chi\) that \(\chi(p)=0\) for each prime \(p\) for which \(\tau_k(p)<\infty\).