Precompact Fréchet topologies on Abelian groups

We study precompact Fréchet topologies on countable Abelian groups. For every countable Abelian group G we introduce the notion of a γG-set and show that there is a precompact Fréchet non-metrizable topology on G if and only if there is an uncountable γG-set that separates points of G. We show that, assuming the existence of an uncountable γ-set, there is a non-metrizable precompact Fréchet topology on every countable Abelian group, and assuming p>ω1, there is a non-metrizable Fréchet topology on every countable group which admits a non-discrete topology at all. We further study the notion of a γG-set and show that the minimal size of a subset of the dual group G⁎ which is not a γG-set is the pseudointersection number p for any countable Abelian group G.