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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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут прикладної математики і механіки НАН України
2014
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/153328 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| 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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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862632991886409728 |
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| author | Banakh, T. Protasov, I. Slobodianiuk, S. |
| author_facet | Banakh, T. Protasov, I. Slobodianiuk, S. |
| 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. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| 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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| first_indexed | 2025-11-30T14:14:44Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-153328 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-11-30T14:14:44Z |
| publishDate | 2014 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | 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 |
| spellingShingle | Densities, submeasures and partitions of groups Banakh, T. Protasov, I. Slobodianiuk, S. |
| title | 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_short | Densities, submeasures and partitions of groups |
| title_sort | densities, submeasures and partitions of groups |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/153328 |
| work_keys_str_mv | AT banakht densitiessubmeasuresandpartitionsofgroups AT protasovi densitiessubmeasuresandpartitionsofgroups AT slobodianiuks densitiessubmeasuresandpartitionsofgroups |