Memory-based and disk-based algorithms for very high degree permutation groups

Abstract

Group membership is a fundamental algorithm, upon which most other algorithms of computational group theory depend. Until now, group membership for permutation groups has been limited to ten million points or less. We extend the applicability of group membership algorithms to permutation groups acting on more than 100,000,000 points. As an example, we experimentally construct a group membership data structure for Thompson's group, acting on 143,127,000 points, in 36 minutes. More significantly, we require approximately 10 GB of RAM for the computation -even though a single permutation of Thompson's group already requires half a gigabyte of storage.In addition, we propose a disk-based group membership algorithm with the promise of extending group membership to well over one billion (1,000,000,000) points. Such a diskbased algorithm has formerly been impossible, due in part to the lack of a practical disk-based algorithm for multiplying and taking inverses of such large permutations. Random access to disk is prohibitively expensive. We demonstrate the first practical disk-based implementation of the basic permutation operations. We also propose a disk-based architecture for group membership data structures.

A parallel architecture for disk-based computing over the Baby Monster and other large finite simple groups

“…Most recently, an implementation by Cooperman and Robinson [20] was able to compute over the Thompson Group, a sporadic simple group acting on 143, 127, 000 points, relatively quickly (36 minutes). The resulting solution could both answer questions of membership and solve for the order of the group.…”

“…Some major achievements have been reached, such as the first constructions of permutation representations and strong generating sets for the Lyons group [15] and Janko's group J4 [19], and now even the Thompson group [38,39,40,20]. These accomplishments have been helped along by the sharing of information at websites such as Wilson's Atlas Web Page [41] which provides initial matrix representations for standard generators.…”

“…Previously permutation representations have been preferred to matrix representations for these large groups due to the the rich and mature body of permutation group algorithms [1,2,3,7,11,12,13,14,20,29,34,36]. However, as the size of the groups worked with increases, a permutation representation becomes less attractive because of the large space requirement to store the permutations.…”

“…This is impractical with today's technology, though it may become feasible (making a direct permutation approach such as Cooperman's and Robinson's [20] feasible as well) once terabyte disks become available. A matrix representation of dimension 4370 over GF (2) is much more practical requiring only 2.3 MB per matrix.…”

“…The algorithm of this paper is an extension of the one used to compute the Thompson Group [20]. However, because it is distributed and out-of-core, we must consider additional factors such as network bandwidth, disk speeds, synchronization, and both memory and disk size limitations.…”