Nearly linear time algorithms for permutation groups with a small base

Abstract

A base of a permutation group G is a subset B of the permutation domain such that only the identity of G fixes B pointwise. The permutation representations of important classes of groups, including all finite simple groups other than the alternating groups, admit O(log n) size bases, where n is the size of the permutation domain. Groups with very small bases dominate the work on permutation groups in much of computational group theory. A series of new combinatorial results allows us to present Monte Carlo algorithms achieving O(n log' n) (c a constant) time and space performance for such groups with respect to the fundamental operations of finding order and testing membership. (The input is a list of generators of the group.) Previous methods have achieved similar space performance only at the expense of increased time performance. Adaptations of a '(cube-doubling" technique [BSZ] and a local expansion property of groups [Ba3] (cf. [Ba4]) are the key to theoretically reducing the time complexity to O(nlogcn). The shared principal novelty of the new ideas is in their abilitv to build and manitmlate certain chains of subsets of a grou~, which are not themselves subgroups, in order to build the point stabilizer subgroup chain. Further combinatorial ideas are used to lower the constant c. Comparative timing estimates, based on asymptotic worst-case analysis, lead us to expect a new implementation to be faster than previous implementations for groups of high degree.