Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2008, Article ID 281734, 5 pages doi:10.1155/2008/281734 Research Article Commutator Length of Finitely Generated Linear Groups Mahboubeh Alizadeh Sanati Department of Sciences, University of Golestan, P.O. Box 49165-386 Gorgan, Golestan, Iran Correspondence should be addressed to Mahboubeh Alizadeh Sanati, malizadeh@gau.ac.ir Received 22 January 2008; Accepted 22 June 2008 Recommended by Nils-Peter Skoruppa 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  12d 3  od 2 , where k is the minimum number of generators of upper triangular subgroup of G and od 2  is a quadratic polynomial in d. Finally we show that if G is a soluble-by-finite group of Pr ¨ uffer rank r then clG ≤ r r  1/2  12r 3  or 2 , where or 2  is a quadratic polynomial in r . Copyright q 2008 Mahboubeh Alizadeh Sanati. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and Motivation We recall that if g is a nonidentity element of the commutator subgroup of an arbitrary nonabelian group G, denote by clg that the least integer such that g can be written as a product of clg  commutators. The commutator length of G is defined as clG sup  clg  | g ∈ G  . 1.1 It is assumed that the commutator length of the identity element is zero. In the present paper, we study clG when G is a soluble-by-finite linear group. In general, the commutator length of a linear group need not be finite. For example, the group G, as a subgroup GL2, Q, generated by  10 21  ,  12 01  1.2 is free and clG is infinite 1. Also, the rank of a group is d if the minimum number of its generators is d. In this case, G is called d-generator, too.