Almost isomorphic torsion-free Abelian groups and similarity of homogeneously decomposable groups. (Q2386455)

We say that two Abelian groups are `almost isomorphic' (`by subgroups with a property \(\mathcal P\)') if each of them is isomorphic to a subgroup of the other group (with the property \(\mathcal P\)). An Abelian group \(A\) is \(\mathcal P\)-`correct' if \(A\cong B\) whenever \(A\) and \(B\) are almost isomorphic (by subgroups with a property \(\mathcal P\)). A general problem can be formulated in the following way: for a property \(\mathcal P\), characterize \(\mathcal P\)-correct groups. We recall here that if \(\mathcal P\) is the property `is a direct summand' then the problem mentioned is one of Kaplansky's test problems. In the first part of the present paper (purely) correct completely decomposable torsion-free groups are studied. The authors present interesting results concerning this topic: A homogeneous completely decomposable group is correct if and only if it is of finite rank or it is a free group (Theorem 3); Every homogeneous completely decomposable group is purely correct (Proposition 6); If a completely decomposable group has a homogeneous component which is not free and has infinite rank then the group is not correct (Theorem 8). In the second part of the paper the authors consider the case: \(\mathcal P\) is the property `is a fully invariant subgroup'. Two technical results are stated in Theorem 10 and Theorem 11. They are used to prove that if a group \(A\) is almost isomorphic by fully invariant subgroups to a group \(G\) which is fully transitively decomposable of finite rank then \(A\) is a homogeneous decomposable group and it is similar to \(G\).