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

## TL;DR

The paper constructs large, torsion-free, countably compact topological groups based on direct sums of rationals, which are $p$-compact without non-trivial convergent sequences, under certain ultrafilter conditions.

## Contribution

It provides the first examples of large, torsion-free, countably compact groups lacking non-trivial convergent sequences, using $p$-compact topologies and selective ultrafilters.

## Key findings

- Existence of $p$-compact group topologies on ${f Q}^{(oldsymbol{rown})}$
- Construction of arbitrarily large countably compact torsion-free groups
- Groups lack non-trivial convergent sequences under specified conditions

## 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.