Commutator Length of Finitely Generated Linear Groups
The commutator length “cl(𝐺)” of a group 𝐺 is the least natural number 𝑐 such that every element of the derived subgroup of 𝐺 is a product of 𝑐 commutators. We give an upper bound for cl(𝐺) when 𝐺 is a 𝑑-generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator...
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| Format: | Article |
| Language: | English |
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Wiley
2008-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2008/281734 |
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| author | Mahboubeh Alizadeh Sanati |
| author_facet | Mahboubeh Alizadeh Sanati |
| author_sort | Mahboubeh Alizadeh Sanati |
| collection | DOAJ |
| description | The commutator length “cl(𝐺)” of a group 𝐺 is the least natural number 𝑐 such that every element of the derived subgroup of 𝐺 is a product of 𝑐 commutators. We give an upper bound for cl(𝐺) when 𝐺 is a 𝑑-generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over 𝐂 that depends only on 𝑑 and the degree of linearity. For such a group 𝐺, we prove that cl(𝐺) is less than
𝑘(𝑘+1)/2+12𝑑3+𝑜(𝑑2), where 𝑘 is the minimum number of generators of (upper) triangular subgroup of 𝐺 and 𝑜(𝑑2) is a quadratic polynomial in 𝑑. Finally we show that if 𝐺 is a
soluble-by-finite group of Prüffer rank 𝑟 then cl(𝐺)≤𝑟(𝑟+1)/2+12𝑟3+𝑜(𝑟2), where 𝑜(𝑟2) is a quadratic polynomial in 𝑟. |
| format | Article |
| id | doaj-art-19de37919fa3405b854d2ee463936279 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2008-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-19de37919fa3405b854d2ee4639362792025-02-03T01:11:50ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252008-01-01200810.1155/2008/281734281734Commutator Length of Finitely Generated Linear GroupsMahboubeh Alizadeh Sanati0Department of Sciences, University of Golestan, P.O. Box 49165-386 Gorgan, Golestan, IranThe commutator length “cl(𝐺)” of a group 𝐺 is the least natural number 𝑐 such that every element of the derived subgroup of 𝐺 is a product of 𝑐 commutators. We give an upper bound for cl(𝐺) when 𝐺 is a 𝑑-generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over 𝐂 that depends only on 𝑑 and the degree of linearity. For such a group 𝐺, we prove that cl(𝐺) is less than 𝑘(𝑘+1)/2+12𝑑3+𝑜(𝑑2), where 𝑘 is the minimum number of generators of (upper) triangular subgroup of 𝐺 and 𝑜(𝑑2) is a quadratic polynomial in 𝑑. Finally we show that if 𝐺 is a soluble-by-finite group of Prüffer rank 𝑟 then cl(𝐺)≤𝑟(𝑟+1)/2+12𝑟3+𝑜(𝑟2), where 𝑜(𝑟2) is a quadratic polynomial in 𝑟.http://dx.doi.org/10.1155/2008/281734 |
| spellingShingle | Mahboubeh Alizadeh Sanati Commutator Length of Finitely Generated Linear Groups International Journal of Mathematics and Mathematical Sciences |
| title | Commutator Length of Finitely Generated Linear Groups |
| title_full | Commutator Length of Finitely Generated Linear Groups |
| title_fullStr | Commutator Length of Finitely Generated Linear Groups |
| title_full_unstemmed | Commutator Length of Finitely Generated Linear Groups |
| title_short | Commutator Length of Finitely Generated Linear Groups |
| title_sort | commutator length of finitely generated linear groups |
| url | http://dx.doi.org/10.1155/2008/281734 |
| work_keys_str_mv | AT mahboubehalizadehsanati commutatorlengthoffinitelygeneratedlineargroups |