Forcing a classification of non-torsion Abelian groups of size at most [formula omitted] with non-trivial convergent sequences

Abstract

We force a classification of all the Abelian groups of cardinality at most 2c that admit a countably compact group with a non-trivial convergent sequence. In particular, we answer (consistently) Question 24 of Dikranjan and Shakhmatov [10] for cardinality at most 2c, by showing that if a non-torsion Abelian group of size at most 2c admits a countably compact Hausdorff group topology, then it admits a countably compact Hausdorff group topology with non-trivial convergent sequences.