On the Bassian property for abelian groups (Q2052359)

The authors call a (non-trivial) abelian group \(G\) Bassian, if there is no injective homomorphism \(G\to G/H\), unless the subgroup \(H=0\). It appears that such a group has to have small rank, for this to happen, so as to limit possibilities for groups \(G/H\). Complete characterization of such groups is obtained by considering separately classes of divisible/reduced groups, torsion groups, torsion-free groups and genuinely mixed groups: A reduced abelian group is Bassian if and only if its torsion-free rank is finite and all the \(p\)-ranks are finite (for all primes \(p\)). A non-reduced abelian group is Bassian, if and only if its divisible part is a direct sum of a finite number of copies of the rationals and its reduced part is Bassian. Some elementary properties of Bassian groups are exhibited.