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We introduce new Monte Carlo methods to speed up and greatly simplify the manipulation of permutation groups. The methods are of a combinatorial character and use elementary group theory only. We achieve a nearly optimal 0(n3 loge n) running time for membership testing, an improvement of two orders of magnitude compared to known elementary algorithms and… (More)

(Received) 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… (More)

A variety of elementary combinatorial techniques for permutation groups are reviewed. It is shown how to apply these techniques to yield faster and/or more space-eecient algorithms for problems including group membership, normal closure, center, base change and Cayley graphs. Emphasis is placed on randomized techniques and new data structures. The paper… (More)

A number of researchers have proposed Cayley graphs and Schreier coset graphs as models for interconnection networks. New algorithms are presented for generating Cayley graphs in a more time-eecient manner than was previously possible. Alternatively, a second algorithm is provided for storing Cayley graphs in a space-eecient manner (log 2 (3) bits per… (More)

A new random base change algorithm is presented for a permutation group <italic>G</italic> acting on <italic>n</italic> points whose worst case asymptotic running time is better for groups with a small to moderate size base than any known deterministic algorithm. To achieve this time bound, the algorithm requires a random generator… (More)

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… (More)

The group membership problem for permutation groups is perhaps the most important problem of computational group theory. Solution of this problem seems to depend intrinsically on constructing a strong generating set. Until now, recognizing if a set of generators is strong has been thought to be as hard as constructing a strong generating set from an… (More)

The construction of point stabilizer subgroups is a problem which has been studied intensively. [1, 4, 5, 10, 11, 12, 14] This work describes a general reduction of certain group constructions to the point stabilizer problem. Examples are given for the centralizer, the normal closure, and a restricted group intersection problem. For the normal closure… (More)

This paper demonstrates the power of the Cayley graph approach to solve speciic applications , such as rearrangement problems and the design of interconnection networks for parallel CPU's. Recent results of the authors for eecient use of Cayley graphs are used here in exploratory analysis to extend recent results of Babai et al. on a family of trivalent… (More)