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Coset enumeration on digital computers

  • John Leech (a1)

In 1936 Todd and Coxeter gave a method ((10), also described in (4), ch. 2) for establishing the order of a finite group defined by a set of relations

satisfied by its generators S1, S2,…, Sk. They enumerate systematically the cosets of a suitable subgroup whose order is evident from the defining relations for the whole group. They describe the method as being ‘purely mechanical’, and since that date the advent of electronic computers has led a number of people to programme the method for automatic execution. Most of this work has remained unpublished; an account is given here of work known to the author.

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(1)Bandler, P. A., M.A. thesis, Manchester, 1956.
(2)Coxeter, H. S. M., The abstract groups Gm,n,υ . Trans. American Math. Soc. 45 (1939), 73150.
(3)Coxeter, H. S. M., The abstract group G3,7,16. Proc. Edinburgh Math. Soc. (2), 13 (1962), 4761.
(4)Coxeter, H. S. M., and Moser, W. O. J., Generators and relations for discrete groups. Ergebnisse der Math. NF, 14 (Springer; Berlin, 1957).
(5)Felsch, H., Programmierung der Restklassenabzählung einer Gruppe nach Untergruppen. Numerische Mathematik, 3 (1961), 250256.
(6)Hall, M., Review of (4). Bull. American Math. Soc. 64 (1958), 106108.
(7)Leech, J., Some definitions of Klein's simple group of order 168 and other groups. Proc. Glasgow Math. Assoc. 5 (1962), 166175.
(7a) Appendix to (7).
(8)Leech, J., and Mennicke, J., Note on a conjecture of Coxeter. Proc. Glasgow Math. Assoc. 5 (1961), 2529.
(9)Maddison, R., Diploma dissertation, University Mathematical Laboratory, Cambridge, 1958.
(10)Todd, J. A., and Coxeter, H. S. M., A practical method for enumerating cosets of a finite abstract group. Proc. Edinburgh Math. Soc. (2), 5 (1936), 2634.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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