Efficient representation of perm groups


This note presents an elementary version of Sims's algorithm for computing strong generators of a given perm group, together with a proof of correctness and some notes about appropriate lowlevel data structures. Upper and lower bounds on the running time are also obtained. (Following a suggestion of Vaughan Pratt, we adopt the convention that perm=permutation, perhaps thereby saving millions of syllables in future research.)

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  1. [1]

    László Babai: “On the length of subgroup chains in the symmetric group,”Communications in Algebra 14 (1986), 1729–1736.

    Google Scholar 

  2. [2]

    László Babai, Eugene M. Luks, and Ákos Seress: “Fast management of permutation groups,”29th Annual Symposium on Foundations of Computer Science (IEEE Computer Society, 1988), 272–282.

  3. [3]

    J. H. Conway: “Three lectures on exceptional groups,” in M. B. Powell and G. Higman, ed.,Finite Simple Groups, Proceedings of the Oxford Instructional Conference on Finite Simple Groups, 1969 (London: Academic Press, 1971), 215–247.

    Google Scholar 

  4. [4]

    Persi Diaconis, R. L. Graham, andWilliam M. Kantor: “The mathematics of perfect shuffles,”Advances in Applied Mathematics 4 (1983), 175–196.

    Google Scholar 

  5. [5]

    Merrick Furst, John Hopcroft, and Eugene Luks: “Polynomial-time algorithms for permutation groups,”21st Annual Symposium on Foundations of Computer Science (IEEE Computer Society, 1980), 36–41.

  6. [6]

    Marshall Hall, Jr. andDavid Wales: “The simple group of order 604,800,”Journal of Algebra 9 (1968), 417–450.

    Google Scholar 

  7. [7]

    Mark Jerrum: “A compact representation for permutation groups,”Journal of Algorithms 7 (1986), 60–78.

    Google Scholar 

  8. [8]

    Charles C. Sims: “Computational methods in the study of permutation groups,” in John Leech, ed.,Computational Problems in Abstract Algebra, Proceedings of a conference held at Oxford University in 1967 (Oxford: Pergamon, 1970), 169–183.

    Google Scholar 

  9. [9]

    Charles C. Sims: “Computation with permutation groups,” in S. R. Petrick, ed.,Proc. Second Symposium on Symbolic and Algebraic Manipulation, Los Angeles, California (New York: ACM, 1971), 23–28.

    Google Scholar 

  10. [10]

    D. E. Taylor: “Pairs of generators for matrix groups,”The Cayley Bulletin 3 (Department of Pure Mathematics, University of Sydney, 1987).

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Dedicated to the memory of Marshall Hall

This research was supported in part by the National Science Foundation under grant CCR-86-10181, and by Office of Naval Research contract N00014-87-K-0502.

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Knuth, D.E. Efficient representation of perm groups. Combinatorica 11, 33–43 (1991). https://doi.org/10.1007/BF01375471

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AMS subject classifications (1991)

  • 20-04
  • 68 Q 25