On the commutator lengths of certain classes of finitely presented groups

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.