Hostname: page-component-745bb68f8f-f46jp Total loading time: 0 Render date: 2025-01-30T01:52:38.876Z Has data issue: false hasContentIssue false
  • English
  • Français

Some results on quotients of triangle groups

Published online by Cambridge University Press:  17 April 2009

Marston D.E. Conder
Marston D.E. Conder
Affiliation:
Department of Mathematics and Statistics, University of Auckland, Private Bag, Auckland, New Zealand.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Given positive integers k, l, m, the (k, l, m) triangle group has presentation δ(k, l, m) = < X, Y, Z | Xk = Yl = Zm = XYZ = 1 >. This paper considers finite permutation representations of such groups. In particular it contains descriptions of graphical and computational techniques for handling them, leading to new results on minimal two-element generation of the finite alternating and symmetric groups and the group of Rubik's cube. Applications to the theory of regular maps and automorphisms of surfaces are also discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 30 , Issue 1 , August 1984 , pp. 73 - 90
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Biggs, N. L. and White, A. T., Permutation Groups and Combinatorial Structures, London Math. Soc. Lecture Note Series, No. 33 (Cambridge University Press, 1979).CrossRefGoogle Scholar
[2]Conder, M. D. E., “Generators for alternating and symmetric groups”, J. London Math. Soc. (2), 22 (1980), 75–86.CrossRefGoogle Scholar
[3]Conder, M. D. E., “More on generators for alternating and symmetric groups”. Quart. J. Math. Oxford Ser. 2, 32 (1981), 137–163.CrossRefGoogle Scholar
[4]Conder, M. D. E., “On the group of Rubik's ‘magic’ cube”, Bull. Inst. Math. Appl., 17 (1981), 241–243.Google Scholar
[5]Coxeter, H. S. M. and Moser, W. O. J., Generators and Relations for Discrete Groups, 4th ed. (Springer-Verlag, 1980).CrossRefGoogle Scholar
[6]Dietze, A. and Schaps, M., “Determining subgroups of a given finite index in a finitely-presented group”, Canadian J. Math. 26 (1974), 769–782.CrossRefGoogle Scholar
[7]Leon, J.S., “On an algorithm for finding a base and strong generating set for a group given by generating permutations”. Math. Camp. 35 (1980), 941–974.CrossRefGoogle Scholar
[8]McKay, J. and Young, K-C., “The non Abelian groups G, |G| < 106 – minimal generating pairs”. Math. Comp. 33 (1979), 812–814 (plus microfiche supplement).Google Scholar
[9]Neumann, P.M., “The SQ-universality of some finitely presented groups”, J. Austral. Math. Soc. 16 (1973), 1–6.CrossRefGoogle Scholar
[10]Singerman, D., “Subgroups of Fuchsian groups and finite permutation groups”, Bull. London Math. Soc. 2 (1970), 319–323.CrossRefGoogle Scholar
[11]Singmaster, D., Notes on Rubik's ‘Magic Cube’ (Polytechnic of the South Bank, London, 1980).Google Scholar
[12]Tucker, T.W., “Finite groups acting on surfaces and the genus of a group”, J. Combin. Theory Ser. B, 34 (1983), 82–98.CrossRefGoogle Scholar
[13]White, A.T., “On the genus of a group”, Trans. Amer. Math. Soc. 173 (1972), 203–214.CrossRefGoogle Scholar
[14]Wielandt, H., Finite Permutation Groups (Academic Press, 1964).Google Scholar