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=A₁ ∪ ⋯ ∪ An of a group G there is a cell Ai of the partition such that G = FAiA⁻¹i for some set F ⊂ G of cardinality |F |≤ n? In this paper we survey several partial solutions of this pro...
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| Veröffentlicht in: | Algebra and Discrete Mathematics |
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| Datum: | 2014 |
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| Sprache: | English |
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Інститут прикладної математики і механіки НАН України
2014
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| Zitieren: | Densities, submeasures and partitions of groups / T. Banakh, I. Protasov, S. Slobodianiuk // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 193–221. — Бібліогр.: 25 назв. — англ. |
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Banakh, T. Protasov, I. Slobodianiuk, S. 2019-06-14T03:21:10Z 2019-06-14T03:21:10Z 2014 Densities, submeasures and partitions of groups / T. Banakh, I. Protasov, S. Slobodianiuk // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 193–221. — Бібліогр.: 25 назв. — англ. 1726-3255 2010 MSC:05E15, 05D10, 28C10. https://nasplib.isofts.kiev.ua/handle/123456789/153328 In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition G=A₁ ∪ ⋯ ∪ An of a group G there is a cell Ai of the partition such that G = FAiA⁻¹i for some set F ⊂ G of cardinality |F |≤ 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 = A₁ ∪ ⋯ ∪ An of a group G there are cells Ai, Aj of the partition such that G = FAjA⁻¹j for some finite set F ⊂ G of cardinality |F| ≤ max₀<k≤n ∑ⁿ⁻kp₌₀kp ≤ n!; G = F ⋅ ⋃x∈ExAiA⁻¹ix⁻¹ for some finite sets F, E ⊂ G with |F| ≤ n; G = FAiA⁻¹iAi for some finite set F ⊂ G of cardinality |F| ≤ n; the set (AiA⁻¹i)⁴ⁿ⁻¹ is a subgroup of index ≤ n in G. The last three statements are derived from the corresponding density results. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Densities, submeasures and partitions of groups Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Densities, submeasures and partitions of groups |
| spellingShingle |
Densities, submeasures and partitions of groups Banakh, T. Protasov, I. Slobodianiuk, S. |
| 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 |
| author |
Banakh, T. Protasov, I. Slobodianiuk, S. |
| author_facet |
Banakh, T. Protasov, I. Slobodianiuk, S. |
| publishDate |
2014 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
In 1995 in Kourovka notebook the second author asked the following problem: is it true that for each partition G=A₁ ∪ ⋯ ∪ An of a group G there is a cell Ai of the partition such that G = FAiA⁻¹i for some set F ⊂ G of cardinality |F |≤ 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 = A₁ ∪ ⋯ ∪ An of a group G there are cells Ai, Aj of the partition such that G = FAjA⁻¹j for some finite set F ⊂ G of cardinality |F| ≤ max₀<k≤n ∑ⁿ⁻kp₌₀kp ≤ n!; G = F ⋅ ⋃x∈ExAiA⁻¹ix⁻¹ for some finite sets F, E ⊂ G with |F| ≤ n; G = FAiA⁻¹iAi for some finite set F ⊂ G of cardinality |F| ≤ n; the set (AiA⁻¹i)⁴ⁿ⁻¹ is a subgroup of index ≤ n in G. The last three statements are derived from the corresponding density results.
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| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/153328 |
| citation_txt |
Densities, submeasures and partitions of groups / T. Banakh, I. Protasov, S. Slobodianiuk // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 193–221. — Бібліогр.: 25 назв. — англ. |
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AT banakht densitiessubmeasuresandpartitionsofgroups AT protasovi densitiessubmeasuresandpartitionsofgroups AT slobodianiuks densitiessubmeasuresandpartitionsofgroups |
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2025-11-30T14:14:44Z |
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2025-11-30T14:14:44Z |
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1850857839238578176 |