Densities, submeasures and partitions of groups
In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition \(G=A_1\cup\dots\cup A_n\) 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...
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| Дата: | 2018 |
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| Мова: | English |
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Lugansk National Taras Shevchenko University
2018
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-10312018-04-26T02:11:00Z Densities, submeasures and partitions of groups Banakh, Taras Protasov, Igor Slobodianiuk, Sergiy partition of a group; density; submeasure; amenable group 05E15, 05D10, 28C10 In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition \(G=A_1\cup\dots\cup A_n\) 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 \(G=A_1\cup\dots\cup A_n\) 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. Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1031 Algebra and Discrete Mathematics; Vol 17, No 2 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1031/554 Copyright (c) 2018 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
partition of a group; density; submeasure; amenable group 05E15 05D10 28C10 |
| spellingShingle |
partition of a group; density; submeasure; amenable group 05E15 05D10 28C10 Banakh, Taras Protasov, Igor Slobodianiuk, Sergiy Densities, submeasures and partitions of groups |
| topic_facet |
partition of a group; density; submeasure; amenable group 05E15 05D10 28C10 |
| format |
Article |
| author |
Banakh, Taras Protasov, Igor Slobodianiuk, Sergiy |
| author_facet |
Banakh, Taras Protasov, Igor Slobodianiuk, Sergiy |
| author_sort |
Banakh, Taras |
| title |
Densities, submeasures and partitions of groups |
| title_short |
Densities, submeasures and partitions of groups |
| title_full |
Densities, submeasures and partitions of groups |
| title_fullStr |
Densities, submeasures and partitions of groups |
| title_full_unstemmed |
Densities, submeasures and partitions of groups |
| title_sort |
densities, submeasures and partitions of groups |
| description |
In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition \(G=A_1\cup\dots\cup A_n\) 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 \(G=A_1\cup\dots\cup A_n\) 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1031 |
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2024-04-12T06:25:06Z |
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2024-04-12T06:25:06Z |
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