Algebraic structure of countably compact non-torsion Abelian groups of size continuum from selective ultrafilters

Assuming the existence of $mathfrak c$ incomparable selective ultrafilters, we classify the non-torsion Abelian groups of cardinality $mathfrak c$ that admit a countably compact group topology. We show that for each $kappa in [mathfrak c, 2^mathfrak c]$ each of these groups has a countably compact group topology of weight $kappa$ without non-trivial convergent sequences and another that has convergent sequences. Assuming the existence of $2^mathfrak c$ selective ultrafilters, there are at least $2^mathfrak c$ non homeomorphic such topologies in each case and we also show that every Abelian group of cardinality at most $2^mathfrak c$ is algebraically countably compact. We also show that it is consistent that every Abelian group of cardinality $mathfrak c$ that admits a countably compact group topology admits a countably compact group topology without non-trivial convergent sequences whose weight has countable cofinality.