Coset enumeration of groups generated by symmetric sets of involutions

The well-known Todd-Coxeter algorithm [14], which may be viewed as a means of constructing permutation representations of finitely presented groups, remains a primary reference for coset enumeration programs. All the strategies and variants of this algorithm perform essentially the same calculations as the original algorithm, merely choosing different orders in which to process the available information. A statement of the basic technique and the early study appear in [7, 8]. A detailed survey and comparison of different strategies are given in [2]. A contemporary work is described in [6, 9]. In more recent work, see [11, 12], we developed two related algorithms for enumerating single and double cosets of any group generated by a finite conjugacy class of involutions. Several finite groups, including all non-abelian finite simple groups, can be symmetrically generated by involutions. Curtis showed how various sporadic simple groups can be so generated, see for instance [4]. The enumerator described in this paper, which can still be viewed as another variant of the Todd-Coxeter algorithm, has substantial improvements (for space, speed, and simplicity of programming) to the version described in [11]. In particular, the additional algebraic information, in the form of coset stabilizing subgroups, is not needed.