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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

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

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.

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Bibliographic databases:
MSC: 05E15, 05D10, 28C10
Revised: 22.04.2014
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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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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
2. O. V. Petrenko, I. V. Protasov, “Ultrafilters on $G$-spaces”, Algebra Discrete Math., 19:2 (2015), 254–269
3. T. Banakh, O. Ravsky, S. Slobodianiuk, “On partitions of $G$-spaces and $G$-lattices”, Int. J. Algebr. Comput., 26:2 (2016), 283–308
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