Pseudocompact group topologies with no infinite compact subsets

We show that every Abelian group satisfying a mild cardi- nal 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 ]). Every pseudocompact Abelian group G with cardinality |G| 22c satisfies this inequality and therefore admits a pseudocompact group topology with property ]. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abelian group admits a pseudocompact group topology with property ]. We also observe that pseudocompact Abelian groups with property ] contain no infinite compact subsets and are examples of Pontryagin re- flexive precompact groups that are not compact.Research supported by the Spanish Ministry of Science (including FEDER funds), grant MTM2008-04599/MTM and Fundaci´o Caixa Castell´o-Bancaixa, grant P1.1B2008-26