On $p$-compact group topologies on direct sums of $\mathbb{Q}$

Abstract

We prove that if $p$ is a selective ultrafilter then ${\mathbb{Q}}^{(\kappa )}$ has a $p$-compact group topology without non-trivial convergent sequences, for each infinite cardinal $\kappa ={\kappa }^{\omega }$. In particular, this gives the first arbitrarily large examples of countably compact groups without non-trivial convergent sequences that are torsion-free.