Abstract. 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-efficient 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 includes both a survey of recent algorithms with which the authors have been associated, and some new algorithms in the same spirit that have not previously appeared in print. Many of the results include both complexity bounds and pseudo-code, along with comments for faster software
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