Fast Group Membership Using a Strong Generating Test for Permutation Groups

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Many important algorithms in computations with permutation groups require an efficient solution to the group membership problem. This requires deciding if a given permutation is an element of a permutation group G specified by a set of generators. Sims [8] developed an elegant solution to this problem. His method relies on the construction of an alternative generating set for G known as a strong generating set which can be easily used to test membership of an arbitrary permutation in G. This algorithm was shown to have worst case time O(n 6). Later versions [1, 5, 6] have improved the theoretical worst case time but without necessarily improving the performance in practice. An algorithm is presented here which has an observed running time of O(n 4) for all permutations groups for which it has been tested. (The worst-case time for this algorithm is O(n 5).) The key idea is a new test [3] for whether a set of generators is a strong generating set. Each call to this last test has worst case time O(n 4). A further reduction in time is achieved by using a fast algorithm for finding reduced generating sets. For groups with small bases, the running running time is O(n 2), which is optimal for the data structure used.