Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-13T08:00:02.465Z Has data issue: false hasContentIssue false
  • English
  • Français

A systematic approach to symmetric presentations. I. Involutory generators

Published online by Cambridge University Press:  24 October 2008

R. T. Curtis ,
A. M. A. Hammas  and
J. N. Bray
R. T. Curtis
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT
A. M. A. Hammas
Affiliation:
Department of Mathematics, University of Medina, Saudi Arabia
J. N. Bray
Affiliation:
School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham B15 2TT
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we conduct a systematic, computerized search for groups generated by small, but highly symmetric, sets of involutions. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. The techniques of symmetric generation developed elsewhere are described afresh, and the results are presented in a convenient tabular form.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 1 , January 1996 , pp. 23 - 34
Copyright
Copyright © Cambridge Philosophical Society 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bray, J. N.. M.Phil. (Sc. Qual.) thesis, Birmingham, in preparation.Google Scholar
[2]Campbell, C. M.. Symmetric presentations and linear groups. Contemp. Math. 45 (1985), 33–39.Google Scholar
[3]Cannon, J. J.. An introduction to the group theory language, Cayley. In Computational Group Theory (Academic Press, 1984), pp. 145–183.Google Scholar
[4]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.. An ATLAS of Finite Groups (Oxford University Press, 1985).Google Scholar
[5]Coxeter, H. S. M. and Moser, W. O. J.. Generators and relations for discrete groups, 4th Ed. (Springer-Verlag, 1984).Google Scholar
[6]Curtis, R. T.. Symmetric Presentations I: Introduction, with particular reference to the Mathieu groups M 12 and M 24, Proceedings of the L M.S. Durham Conference on ‘Groups and Combinatorics’ (1990).Google Scholar
[7]Curtis, R. T.. Symmetric Presentations II: The Janko group J1. J. London Math. Soc. (2) 47 (1993), 294–308.Google Scholar
[8]Curtis, R. T.. Geometric interpretation of the ‘natural’ generators of the Mathieu groups. Math. Proc. Cam. Phil. Soc. 107 (1990), 19–26.Google Scholar
[9]Dickson, L. E.. Linear groups, with an exposition of the Galois field theory (Leipzig, 1901).Google Scholar
[10]Hammas, A. M. A.. Symmetric presentations of some finite groups, Ph.D. thesis, Birmingham (1991).Google Scholar
[11]Jabbar, A.. Symmetric presentations of subgroups of the Conway groups and related topics. Ph.D. thesis, Birmingham (1992).Google Scholar
[12]Johnson, D. L.. Presentations of groups. London Math. Soc. Lecture Notes 22 (Cambridge University Press, 1976).Google Scholar
[13]Robertson, E. F. and Campbell, C. M.. Symmetric presentations. In Proceedings of the 1987 Singapore conference (Walter de Gruyter, 1989), pp. 497–506.Google Scholar