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

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