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For a finite groupG=〈X〉 (X≠G), the least positive integerMLX(G)is called the maximum length ofGwith respect to the generating setXif every element ofGmay be represented as a product of at mostMLX(G)elements ofX. The maximum length ofG, denoted byML(G), is defined to be the minimum of{MLX(G)|G=〈X〉,X≠G,X≠G−{1G}}. The well-known commutator length of a groupG, denoted byc(G), satisfies the inequalityc(G)≤ML(G′), whereG′is the derived subgroup ofG. In this paper we study the properties ofML(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.

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