International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 74981, 9 pages
doi:10.1155/IJMMS/2006/74981
On the commutator lengths of certain classes of finitely
presented groups
1Mathematics Department, Teacher Training University, 49 Mofateh Avenue, Tehran 15614, Iran
2Institute of Mathematics, St. Andrews University, St. Andrews, Scotland KY16 9SS, UK
Received 23 June 2004; Revised 4 July 2005; Accepted 28 March 2006
Copyright © 2006 H. Doostie and P. P. Campbell. 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
For a finite group G=〈X〉 (X≠G), the least
positive integer MLX(G) is called the maximum length of G
with respect to the generating set X if every element of G may
be represented as a product of at most MLX(G) elements of X.
The maximum length of G, denoted by ML(G), is defined to be
the minimum of {MLX(G)|G=〈X〉,X≠G,X≠G−{1G}}. The well-known commutator length of a group
G, denoted by c(G), satisfies the inequality c(G)≤ML(G′), where G′ is the derived subgroup of G. In this paper
we study the properties of ML(G) and by using this inequality
we give upper bounds for the commutator lengths of certain classes
of finite groups. In some cases these upper bounds involve the
interesting sequences of Fibonacci and Lucas numbers.