International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 281734, 5 pages
Commutator Length of Finitely Generated Linear Groups
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 “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -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 that depends only on and the degree of linearity. For such a group , we prove that is less than
, where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a
soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .