Pseudocompact Group Topologies with No Infinite Compact Subsets

We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property ♯). This criterion is used in conjunction with an analysis of the algebraic structure of pseudocompact groups to obtain, under the Generalized Continuum Hypothesis (GCH), a characterization of those pseudocompact groups that admit such a topology. We prove in particular that each of the following groups admits a pseudocompact group topology with property ♯: – Pseudocompact groups of cardinality not greater than 2 c . – (GCH) Connected pseudocompact groups. – (GCH) Pseudocompact groups whose torsion-free rank has uncountable cofinality. We also observe that pseudocompact groups with property ♯ contain no infinite compact subsets and are examples of Pontryagin reflexive precompact groups that are not compact.