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

H. Doostie1 and P. P. Campbell2

 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

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.