A complete description of the algebraic structure of pseudocompact abelian groups

Abstract

We solve a major open problem from [D. Dikranjan, D. Shakhmatov, Algebraic structure of pseudocompact groups, Memoirs AMS 633 (1998)] by describing abelian groups which admit a pseudocompact Hausdorff group topology. First, we prove that an abelian group G can be equipped with a pseudocompact Hausdorff group topology iff it can be decomposed into a direct sum G=B\oplus H of a bounded torsion group B and agroup H satisfying |nH|=|H| for all positive integers $n$, such that both B and H admit a pseudocompact Hausdorff group topology. Second, for an unbounded abelian group G, we denote by m the minimal positive integer n such that |nG|=|kG| for all integers k\ge n. Then G admits a pseudocompact group topology iff both G[m]={g\in G: mg=0} and mG\cong G/G[m] admit a pseudocompact Hausdorff group topology.