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SSGP topologies on abelian groups of positive finite divisible rank

Volume 244 / 2019

Dmitri Shakhmatov, Víctor Hugo Yañez Fundamenta Mathematicae 244 (2019), 125-145 MSC: Primary 22A05; Secondary 03E99, 06A06, 20K27, 54E35, 54H11. DOI: 10.4064/fm463-3-2018 Published online: 27 September 2018

Abstract

For a subset $A$ of a group $G$, we denote by $\def\grp#1{\langle{#1}\rangle}\grp{A}$ the smallest subgroup of $G$ containing $A$ and let $\def\Cyc{\mathrm{Cyc}}\Cyc(A)=\{x\in G: \def\grp#1{\langle{#1}\rangle}\grp{\{x\}}\subseteq A\}$. A topological group $G$ is SSGP if $\def\grp#1{\langle{#1}\rangle}\grp{\def\Cyc{\mathrm{Cyc}}\Cyc(U)}$ is dense in $G$ for every neighbourhood $U$ of the identity of $G$. The SSGP groups form a proper subclass of the class of minimally almost periodic groups.

Comfort and Gould asked about a characterization of abelian groups which admit an SSGP group topology. An “almost complete” characterization was found by Dikranjan and the first author. The remaining case is resolved here. As a corollary, we give a positive answer to another question of Comfort and Gould by showing that if an abelian group admits an SSGP($n$) group topology for some positive integer $n$, then it admits an SSGP group topology as well.

Authors

  • Dmitri ShakhmatovDivision of Mathematics, Physics
    and Earth Sciences
    Graduate School of Science
    and Engineering
    Ehime University
    Matsuyama 790-8577, Japan
    e-mail
  • Víctor Hugo YañezDoctor’s Course
    Graduate School of Science
    and Engineering
    Ehime University
    Matsuyama 790-8577, Japan
    e-mail

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