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

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