In this chapter, we start the main topic of this book with an overview of permutation group algorithms.

Polynomial-Time Algorithms

In theoretical computer science, a universally accepted measure of efficiency is polynomial-time computation. In the case of permutation group algorithms, groups are input by a list of generators. Given G = 〈S〉 ≤ Sn, the input is of length |S|n and a polynomial-time algorithm should run in O((|S|n)c) for some fixed constant c. In practice, |S| is usually small: Many interesting groups, including all finite simple groups, can be generated by two elements, and it is rare that in a practical computation a permutation group is given by more than ten generators. On the theoretical side, any G ≤ Sn can be generated by at most n/2 permutations (cf. [McIver and Neumann, 1987]). Moreover, any generating set S can be easily reduced to less than n2 generators in O(|S|n2) time by a deterministic algorithm (cf. Exercise 4.1), and in Theorem 10.1.3 we shall describe how to construct at most n – 1 generators for any G ≤ Sn. Hence, we require that the running time of a polynomial-time algorithm is O(nc + |S|n2) for some constant c.

In this book, we promote a slightly different measure of complexity involving n, |S|, and log |G| (cf. Section 3.2), which better reflects the practical performance of permutation group algorithms.