On the commutator lengths of certain classes of finitely presented groups

For a finite group G = 〈 X 〉 ( X ≠ G ) , the leastpositive integer ML X ( G ) is called the maximum length of G with respect to the generating set X if every element of G maybe represented as a product of at most ML X ( G ) elements of X .The maximum length of G , denoted by ML ( G ) , is defined to bethe minimum of { ML X ( G ) | G = 〈 X 〉 , X ≠ G , X ≠ G − { 1 G } } . 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 paperwe study the properties of ML ( G ) and by using this inequalitywe give upper bounds for the commutator lengths of certain classesof finite groups. In some cases these upper bounds involve theinteresting sequences of Fibonacci and Lucas numbers.