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Algebra Discrete Math., 2014, Volume 17, Issue 2, Pages 193–221 (Mi adm466)  

This article is cited in 3 scientific papers (total in 3 papers)

SURVEY ARTICLE

Densities, submeasures and partitions of groups

Taras Banakhab, Igor Protasovc, Sergiy Slobodianiukc

a Ivan Franko National University of Lviv, Ukraine
b Jan Kochanowski University in Kielce, Poland
c Taras Shevchenko National University, Kyiv, Ukraine

Abstract: In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition $G=A_1\cup…\cup A_n$ of a group $G$ there is a cell $A_i$ of the partition such that $G=FA_iA_i^{-1}$ for some set $F\subset G$ of cardinality $|F|\le n$? In this paper we survey several partial solutions of this problem, in particular those involving certain canonical invariant densities and submeasures on groups. In particular, we show that for any partition $G=A_1\cup…\cup A_n$ of a group $G$ there are cells $A_i$, $A_j$ of the partition such that
  • $G=FA_jA_j^{-1}$ for some finite set $F\subset G$ of cardinality $|F|\le \max_{0<k\le n}\sum_{p=0}^{n-k}k^p\le n!$;
  • $G=F\cdot\bigcup_{x\in E}xA_iA_i^{-1}x^{-1}$ for some finite sets $F,E\subset G$ with $|F|\le n$;
  • $G=FA_iA_i^{-1}A_i$ for some finite set $F\subset G$ of cardinality $|F|\le n$;
  • the set $(A_iA_i^{-1})^{4^{n-1}}$ is a subgroup of index $\le n$ in $G$.
The last three statements are derived from the corresponding density results.

Keywords: partition of a group; density; submeasure; amenable group.

Full text: PDF file (296 kB)
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Bibliographic databases:
MSC: 05E15, 05D10, 28C10
Received: 22.04.2014
Revised: 22.04.2014
Language:

Citation: Taras Banakh, Igor Protasov, Sergiy Slobodianiuk, “Densities, submeasures and partitions of groups”, Algebra Discrete Math., 17:2 (2014), 193–221

Citation in format AMSBIB
\Bibitem{BanProSlo14}
\by Taras~Banakh, Igor~Protasov, Sergiy~Slobodianiuk
\paper Densities, submeasures and partitions of groups
\jour Algebra Discrete Math.
\yr 2014
\vol 17
\issue 2
\pages 193--221
\mathnet{http://mi.mathnet.ru/adm466}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3287929}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Protasov I., Slobodianiuk S., “Partitions of Groups Into Large Subsets”, J. Group Theory, 18:2 (2015), 291–298  crossref  mathscinet  zmath  isi  elib  scopus
    2. O. V. Petrenko, I. V. Protasov, “Ultrafilters on $G$-spaces”, Algebra Discrete Math., 19:2 (2015), 254–269  mathnet  mathscinet
    3. T. Banakh, O. Ravsky, S. Slobodianiuk, “On partitions of $G$-spaces and $G$-lattices”, Int. J. Algebr. Comput., 26:2 (2016), 283–308  crossref  mathscinet  zmath  isi  elib  scopus
  • Algebra and Discrete Mathematics
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