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We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in O (k^2/3). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Omega(k^2/3), we give a reduction from a specialarXiv:quant-ph/0506265v4 fatcat:sfknwj2apbdrfpamh2u5iis4mm