Double-coset enumeration algorithm for symmetrically generated groups

The Todd-Coxeter algorithm described in [13] remains a primary reference for coset enumeration programs. It may be viewed as a means of constructing permutation representations of finitely presented groups. A number of effective computer programs for singlecoset enumeration have been described, see, for example, [2, 7, 8]. Enumerating double cosets, rather than single cosets, gives substantial reduction in total cosets defined which leads to minimizing the storage (and time) needed. In [8, 9] Linton has developed two double-coset enumeration programs. The process in the earlier one, which is a corrected version of the algorithm proposed in [3], is viewed as a direct generalization of the ordinary single-coset enumeration. The later one is related to the present algorithm but with different calculations. In this paper, we present a double-coset enumeration algorithm which is specially developed for groups symmetrically generated by involutions. Several finite groups, including all nonabelian finite simple groups, can be generated by symmetric sets of involutions, see, for example, [6, 10, 11]. The algorithm has been implemented as a Magma [1] program and this implementation has been used with success to check symmetric presentations for many finite simple groups.