The Schreier-Sims algorithm

A base and strong generating set provides an effective computer representation for a permutation group. This representation helps us to calculate the group order, list the group elements, generate random elements, test for group membership and store group elements efficiently. Knowledge of a base and strong generating set essentially reduces these tasks to the calculation of certain orbits. Given an arbitrary generating set for a permutation group, the SchreierSims algorithm calculates a base and strong generating set. We describe several variations of this method, including the Todd-Coxeter, random and extending Schreier-Sims algorithms. Matrix groups over finite fields can also be represented by a base and strong generating set, by considering their action on the underlying vector space. A practical implementation of the random Schreier-Sims algorithm for matrix groups is described. We investigate the effectiveness of this implementation for computing with soluble groups, almost simple groups, simple groups and related constructions. We consider in detail several aspects of the implementation of the random Schreier-Sims algorithm. In particular, we examine the generation of random group elements and choice of “stopping condition”. The main difficulty in applying this algorithm to matrix groups is that the orbits which must be calculated are often very large. Shorter orbits can be found by extending the point set to include certain subspaces of the underlying vector space. We demonstrate that even greater improvements in the performance of the random Schreier-Sims algorithm can be achieved by using the orbits of eigenvectors and eigenspaces of the generators of the group.