Strong generating set

Abstract: We present new algorithms for permutation group manipulation. Our methods result in an improvement of nearly an order of magnitude in the worst-case analysis for the fundamental problems of finding strong generating sets and testing membership. The normal structure of the group is brought into play even for such elementary issues. An essential element is the recognition of large alternating composition factors of the given group and subsequent extension of the permutation domain to display the natural action of these alternating groups. Further new features include a novel fast handling of alternating groups and the sifting of defining relations in order to link these and other analyzed factors with the rest of the group. The analysis of the algorithm depends on the classification of finite simple groups. In a sequel to this paper, using an enhancement of the present method, we shall achieve a further order of magnitude improvement.

Abstract: The computation of a strong generating set for a permutation group acting on a set $\Omega $ of n points is the fundamental operation that underlies most of the algorithms in computational group theory. Sims gave a change of basis algorithm that transforms a strong generating set relative to one ordering of $\Omega $ into a strong generating set relative to a different ordering. Base change is crucial for many of the important algorithms that have been implemented in the Cayley system, and is also important for many applications of computational group theory to combinatorial and search problems. Sims’s base change has worst-case time $O(n^5 )$. The main result of this paper is a new change of basis algorithm that has worst-case time $O(n^3 )$.

Abstract: A new random base change algorithm is presented for a permutation group G acting on n points whose worst case asymptotic running time is better for groups with a small to moderate size base than any known deterministic algorithm. To achieve this time bound, the algorithm requires a random generator Rand(G) producing a random element of G with the uniform distribution and so that each call to Rand(G) takes time O(log(|G|)n). The random base change algorithm has probability 1 -- 1/|G|2 of completing in time O(log2(|G|)n) and outputting a data structure for representing the point stabilizer sequence relative to the new ordering which requires O(log(|G|)n) space and which can be used to test group membership in time O,(log(|G|)n). The time to build a data structure for computing a Rand(G) with the above properties from a strong generating set for G is dominated by the time to construct the strong generating set from the original set of generators.