Fast Group Membership Using a Strong Generating Test for Permutation Groups

Abstract

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.

The work of this author was supported in part by the National Science Foundation under grant number DCR-8603293.