A group G of permutations of n elements which is generated by a subset S is said to have the diameter d relative to S if every permutation in G can be expressed as a product of at most d permutations from S. It is known that the computation of d is NP-hard in the general case. It is also known that d=O(n²) if every generator is a cycle moving at most k elements. This paper gives a short proof that d=O(n^k) if each generator moves at most k elements, but without cyclicity restriction.