Mahboubeh Alizadeh Sanati

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.

Abstract

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.