Forcing a classification of non-torsion Abelian groups of size at most $2^\mathfrak c$ with non-trivial convergent sequences

TL;DR

This paper classifies non-torsion Abelian groups of size up to $2^\mathfrak c$ that can have a countably compact topology with non-trivial convergent sequences, answering a specific open question in the field.

Contribution

It provides a classification result for Abelian groups of certain size admitting specific topological structures, extending previous understanding and resolving an open question.

Findings

Non-torsion Abelian groups of size at most $2^\mathfrak c$ admit countably compact topologies with non-trivial convergent sequences.

Answer to Question 24 of Dikranjan and Shakhmatov is affirmative for groups of size at most $2^\mathfrak c$.

Classification of such groups under the given conditions is achieved.

Abstract

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