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.
A parallel variamt of breadth-first search for distributed computing is presented. The variant allows exhaustive enumeration of elements of a search space (implicitly defined graph) in which the representation of all graph nodes would otherwise require more than the total available memory. This algorithm requires the use of a tadpole data structure to partition the sezwch space into connected subgraphs, with each subgraph stored within the memory of a single processor. Thus, the graph of the nodes are stored in a way that adds greater spatial locidity, thereby reducing communication among processors. The algorithm enumerates the tadpoles in a breadth-first manner while executing depth-first search within each tadpole. The search within each tadpole is reduced to tree search. A parameter is defined that allows a linear tradeoff in which memory and communication are each reduced linearly as CPU time grows. The result appears to fill a gap in the literature. which has concentrated on parallel depth-first search, but has been relatively sparse in the case of parallel breadth-first search.
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