International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 281734, 5 pages
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

Received 22 January 2008; Accepted 22 June 2008

Academic Editor: Nils-Peter Skoruppa

Copyright © 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.


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.