Co-minimal abelian groups

An abelian group A was called minimal in [3], if A is isomorphic to all its subgroups of finite index. We study the dual notion and call A cominimal if A is isomorphic to A/K for all finite subgroups K of A. We will see that minimal and co-minimal groups exhibit a similar behavior in some cases, but there are several differences. While a reduced p-group A is minimal if and only if A/p ^ωA is minimal, this no longer holds for cominimal p-groups. We show that a separable p-group A is co-minimal if and only if A is minimal, This does not hold for p-groups with elements of infinite height. We find necessary conditions for co-minimal p-groups in terms of their Ulm-Kaplansky invariants, which are also sufficient for totally projective p-groups. If A is a mixed group with a knice system, also known as Axiom 3 modules, then A is co-minimal if and only if t(A), the torsion part of A, is co-minimal. We construct an example of a mixed group A such that t(A) is a totally projective p-group of length ω + 1 such that t(A) is co-minimal but A is not co-minimal. Moreover, we construct p-groups G of length ω + 1 such that all Ulm-Kaplansky invariants of G are infinite, i.e. G is minimal, but G is not co-minimal.