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# Journal of Group Theory

Editor-in-Chief: Parker, Christopher W. / Wilson, John S.

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Volume 20, Issue 1 (Jan 2017)

# A note on Factoring groups into dense subsets

Igor Protasov
• Department of Cybernetics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine
• Email:
/ Serhii Slobodianiuk
• Department of Mechanics and Mathematics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine
• Email:
Published Online: 2016-05-18 | DOI: https://doi.org/10.1515/jgth-2016-0021

## Abstract

Let G be a group of cardinality $\kappa >{\mathrm{\aleph }}_{0}$ endowed with a topology $\mathcal{𝒯}$ such that $|U|=\kappa$ for every non-empty $U\in \mathcal{𝒯}$ and $\mathcal{𝒯}$ has a base of cardinality κ. We prove that G can be factorized $G=AB$ (i.e. each $g\in G$ has a unique representation $g=ab$, $a\in A$, $b\in B$) into dense subsets A, B, $|A|=|B|=\kappa$. We do not know if this statement holds for $\kappa ={\mathrm{\aleph }}_{0}$ even if G is a topological group.

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## About the article

Revised: 2016-04-29

Published Online: 2016-05-18

Published in Print: 2017-01-01

Citation Information: Journal of Group Theory, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, Export Citation