Generalisations of the Todd-Coxeter Algorithm

  • S. A. Linton
Part of the Mathematics and Its Applications book series (MAIA, volume 325)


The Todd-Coxeter coset enumeration algorithm was perhaps the first non-trivial algorithm from pure mathematics to be programmed for a digital computer. Recently the author has developed two related algorithms, the double coset enumeration algorithm and the vector enumeration algorithm. This paper establishes a common framework for the three algorithms, based on a related algorithm for constructing transformation representations of monoids.


Regular Representation Free Representation Double Coset Enumeration Algorithm Permutation Representation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    J. J. Cannon, L. A. Dimino, G. Havas, and J. M. Watson, Implementation and Analysis of the Todd—Coxeter Algorithm, Math. Comp. 27 (1973), 463–490.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    S. Carmody, and R. F. C. Walters, The Todd—Coxeter Procedure and Left Kan Extensions,this volume, pp. 53–76.Google Scholar
  3. [3]
    G. Havas, Coset Enumeration Strategies, in: S. M. Watt (ed.), (Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, New York: ACM Press, 1991, pp. 191–199.CrossRefGoogle Scholar
  4. [4]
    J. Leech, Coset Enumeration on Digital Computers, Proc. Camb. Phil. Soc. 59 (1963), 257–267.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    J. Leech, Coset Enumeration, in: M. D. Atkinson (ed.), Computational Group Theory, New York: Academic Press, 1984, pp. 3–18.Google Scholar
  6. [6]
    G. Labonté, An Algorithm for the Construction of Matrix Representations for Finitely Presented Non-commutative Algebras, J. Symbolic Comput. 9 (1990), 27–38.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    S. A. Linton, Double Coset Enumeration, J. Symbolic Comput. 12 (1991), 415–426.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S. A. Linton, Constructing Matrix Representations of Finitely Presented Groups, J. Symbolic Comput. 12 (1991), 427–438.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    S. A. Linton, On Vector Enumeration, Linear Algebra and its Applications 192 (1993), 235–248.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    L. H. Soicher, private communication.Google Scholar
  11. [11]
    E. F. Robertson, Programs to enumerate semigroups and using these programs to study semigroup presentations, talk delivered at the Oberwolfach workshop on Computational Group Theory, 1992.Google Scholar
  12. [12]
    J. A. Todd, H. S. M. Coxeter, A Practical Method for Enumerating Cosets of a Finite Abstract Group, Proc. Edinburgh Math. Soc. 5 (1936), 26–34.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • S. A. Linton
    • 1
  1. 1.School of Mathematical and Computational SciencesUniversity of St. AndrewsSt Andrews, FifeScotland

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