Constructing permutation representations for
                    matrix groups in parallel environments

Tselman, Michael

doi:10.1090/dimacs/028/20

Cited by 1 publication

(2 citation statements)

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“…It is reported in Tselman (1996) and briefly summarized here. Two implementations were done: in the C language on the MasPar MP-1, a SIMD (single instruction, multiple data); and in LISP on a cluster of SPARC-2 workstations using LISP/MPI, a package described in .…”

Section: Parallel Implementationmentioning

confidence: 99%

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Constructing Permutation Representations for Matrix Groups

Cooperman

Finkelstein

Tselman

et al. 1997

Journal of Symbolic Computation

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New techniques, both theoretical and practical, are presented for constructing permutation representations for computing with matrix groups defined over finite fields. The permutation representation is constructed on a conjugacy class of subgroups of prime order. We construct a base for the permutation representation, which in turn simplifies the computation of a strong generating set. In addition, we present an elementary test for checking the simplicity of the permutation image.The theory has been successfully tested on a representation of the sporadic simple group Ly, discovered by Lyons (1972). With no a priori assumptions, we find a permutation representation of degree 9 606 125 on a conjugacy class of subgroups of order 3, find the order of the resulting permutation group, and verify simplicity. A Monte Carlo variation of the algorithm was used to achieve better space and time efficiency. The construction of the permutation representation required four CPU days on a SPARCserver 670MP with 64 MB. The permutation representation was used implicitly in the sense that the group element was stored as a matrix, and its permutation action on a "point" was determined using a pre-computed data structure. Thus, additional computations required little additional space. The algorithm has also been implemented using the MasPar MP-1 SIMD parallel computer and 8 SPARC-2's running under MPI. The results of those parallel experiments are briefly reviewed.

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Section: Parallel Implementationmentioning

confidence: 99%

“…It also required a new heuristic described in Tselman (1996) in order to avoid the expense of resolving hash collisions among distributed parallel processors. This heuristic does not come with a guarantee of correctness, but in the experiments, the heuristic never failed.…”

Section: Parallel Implementationmentioning

confidence: 99%