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.
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BibTeX
@misc{dikranjan2026complete,
title = {{A complete description of the algebraic structure of pseudocompact abelian groups}},
author = {Dikran Dikranjan and Alejandro Dorantes-Aldama and Dmitri Shakhmatov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03270},
eprint = {2611.03270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03270},
}
RIS
TY - UNPB
TI - A complete description of the algebraic structure of pseudocompact abelian groups
AU - Dikran Dikranjan
AU - Alejandro Dorantes-Aldama
AU - Dmitri Shakhmatov
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03270
UR - https://arxiv.org/abs/2611.03270
AB - 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.
ER -