Constructing permutation representations for large matrix groups

New techniques, both theoretical and practical, are presented for constructing a permutation representation for a matrix group. We assume that the resulting permutation degree, n, can be 10,000,000 and larger. The key idea is to build the new permutation representation using the conjugation action on a conjugacy class of subgroups of prime order. A unique signature for each group element corresponding to the conjugacy class is used in order to avoid matrix multiplication. The requirement of at least n matrix multiplications would otherwise have made the computation hopelessly impractical. Additional software optimizations are described, which reduce the CPU time by at least an additional factor of 10. Further, a special data structure is designed that serves both as a search tree and as a hash array, while requiring space of only 1.6n log2 n bits. The technique has been implemented and tested on the sporadic simple group Ly, discovered by Lyons [9], in both a sequential (SPARCserver 670MP) and parallel SIMD (MasPar MP-1) version. Starting with a generating set for Ly as a subgroup of GL(111,5) [5], a set of generating permutations for Ly acting on 9, 606, 125 points is constructed as well as a base for this permutation representation. The sequential version required four days of CPU time to construct a data structure which can be used to compute the permutation image of an arbitrary matrix. The parallel version did so in 12 hours. Work is in progress on a faster parallel implementation.

[1]  G. Seitz,et al.  On the minimal degrees of projective representations of the finite Chevalley groups , 1974 .

[2]  Daniel Gorenstein.  The Classification of Finite Simple Groups , 1983 .

[3]  László Babai,et al.  Nearly linear time algorithms for permutation groups with a small base , 1991, ISSAC '91.

[4]  Gene Cooperman,et al.  Combinatorial Tools for Computational Group Theory , 1991, Groups And Computation.

[5]  Gregory Butler,et al.  The Schreier algorithm for matrix groups , 1976, SYMSAC '76.

[6]  J. Conway,et al.  ATLAS of Finite Groups , 1985 .

[7]  László Babai,et al.  Las Vegas algorithms for matrix groups , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[8]  Richard Lyons,et al.  Evidence for a new finite simple group , 1972 .

[9]  László Babai,et al.  Local expansion of vertex-transitive graphs and random generation in finite groups , 1991, STOC '91.

[10]  Charles C. Sims.  The Existence and Uniqueness of Lyons' Group , 1973 .

[11]  Charles C. Sims,et al.  Computation with permutation groups , 1971, SYMSAC '71.

[12]  Derek F. Holt,et al.  Testing modules for irreducibility , 1994, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[13]  Gregory Butler,et al.  Cayley, Version 4: The User Language , 1988, ISSAC.

[14]  Steve Linton.  The Art and Science of Computing in Large Groups , 1995 .

[15]  B. Huppert.  Endliche Gruppen I , 1967 .

[16]  Eugene M. Luks,et al.  Computing in solvable matrix groups , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[17]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[18]  Endre Szemerédi,et al.  On the Complexity of Matrix Group Problems I , 1984, FOCS.

[19]  John J. Cannon,et al.  Computing in permutation and matrix groups. I. Normal closure, commutator subgroups, series , 1982 .

[20]  Robert Beals,et al.  Computing Blocks of Imprimitivity for Small-Base Groups in Nearly Linear Time 17 , 1991, Groups And Computation.

[21]  Gene Cooperman,et al.  STAR/MPI: binding a parallel library to interactive symbolic algebra systems , 1995, ISSAC '95.

[22]  Gene Cooperman,et al.  Computing with matrix groups using permutation representations , 1995, ISSAC '95.