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.