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 |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |