Commutator Length of Finitely Generated Linear Groups

Mahboubeh Alizadeh Sanati

International Journal of Mathematics and Mathematical Sciences, 2008, vol. 2008, 1-5

Abstract:

The commutator length “ c l ( ð º ) ” of a group ð º is the least natural number ð ‘ such that every element of the derived subgroup of ð º is a product of ð ‘ commutators. We give an upper bound for c l ( ð º ) when ð º is a ð ‘‘ -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 ð ‚ that depends only on ð ‘‘ and the degree of linearity. For such a group ð º , we prove that c l ( ð º ) is less than 𠑘 ( 𠑘 + 1 ) / 2 + 1 2 ð ‘‘ 3 + ð ‘œ ( ð ‘‘ 2 ) , where 𠑘 is the minimum number of generators of (upper) triangular subgroup of ð º and ð ‘œ ( ð ‘‘ 2 ) is a quadratic polynomial in ð ‘‘ . Finally we show that if ð º is a soluble-by-finite group of Prüffer rank ð ‘Ÿ then c l ( ð º ) ≤ ð ‘Ÿ ( ð ‘Ÿ + 1 ) / 2 + 1 2 ð ‘Ÿ 3 + ð ‘œ ( ð ‘Ÿ 2 ) , where ð ‘œ ( ð ‘Ÿ 2 ) is a quadratic polynomial in ð ‘Ÿ .

Date: 2008
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:281734

DOI: 10.1155/2008/281734

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