Hostname: page-component-745bb68f8f-s22k5 Total loading time: 0 Render date: 2025-01-11T15:39:05.716Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructing maximal subgroups of orthogonal groups

Part of: Linear algebraic groups and related topics Theory of computing Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  01 May 2010

Derek F. Holt  and
Colva M. Roney-Dougal
Derek F. Holt
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom (email: d.f.holt@warwick.ac.uk)
Colva M. Roney-Dougal
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, United Kingdom (email: colva@mcs.st-and.ac.uk)

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.

In this paper we construct the maximal subgroups of geometric type of the orthogonal groups in dimension d over GF(q) in O(d3+d2log q+log qlog log q) finite field operations.

MSC classification

Secondary: 20D06: Simple groups 20E28: Maximal subgroups 20G40: Linear algebraic groups over finite fields 68Q25: Analysis of algorithms and problem complexity
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 164 - 191
Copyright
Copyright © London Mathematical Society 2010

References

[1] Aschbacher, M., ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984) 469–514.Google Scholar
[2] Bosma, W. and Cannon (eds), J. J., Handbook of Magma functions, version 2.14 (2007)http://magma.maths.usyd.edu.au/magma/.Google Scholar
[3] Bürgisser, P., Clausen, M. and Shokrollahi, M. A., Algebraic complexity theory (Springer, Berlin, 1997).CrossRefGoogle Scholar
[4] Cannon, J. J. and Holt, D. F., ‘Computing maximal subgroups of finite groups’, J. Symbol. Comput. 37 (2004) 589–609.CrossRefGoogle Scholar
[5] Coppersmith, D. and Winograd, S., ‘Matrix multiplication via arithmetic progressions’, J. Symbol. Comput. 9 (1990) 251–280.CrossRefGoogle Scholar
[6] Dye, R. H., ‘A geometric characterization of the special orthogonal groups and the Dickson invariant’, J. London Math. Soc. 15 (1977) 472–476.CrossRefGoogle Scholar
[7] Eick, B. and Hulpke, A., ‘Computing the maximal subgroups of a permutation group I’, Groups and computation III, Ohio State University Research Institute Publications 8 (eds Kantor, W. M. and Seress, À.; de Gruyter, Berlin, 2001) 155–168.Google Scholar
[8] Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 5th edn (Oxford University Press, Oxford, 1979).Google Scholar
[9] Hiss, G. and Malle, G., ‘Low dimensional representations of quasi-simple groups’, LMS J. Comput. Math. 4 (2001) 22–63 [Corrigenda: LMS J. Comput. Math. 5 (2002) 95–126].CrossRefGoogle Scholar
[10] Holt, D. F. and Roney-Dougal, C. M., ‘Constructing maximal subgroups of classical groups’, LMS J. Comput. Math. 8 (2005) 46–79.Google Scholar
[11] Howlett, R. B., Rylands, L. J. and Taylor, D. E., ‘Matrix generators for exceptional groups of Lie type’, J. Symbol. Comput. 31 (2001) 429–445.CrossRefGoogle Scholar
[12] Kantor, W. M. and Seress, Á., ‘Black box classical groups’, Mem. Amer. Math. Soc. 149 (2001).Google Scholar
[13] Kleidman, P. B., ‘The maximal subgroups of the finite 8-dimensional orthogonal groups PΩ +8(q) and of their automorphism groups’, J. Algebra 110 (1987) 173–242.CrossRefGoogle Scholar
[14] Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[15] Lübeck, F., ‘Small degree representations of finite Chevalley groups in defining characteristic’, LMS J. Comput. Math. 4 (2001) 135–169.Google Scholar
[16] Murray, S. H. and Roney-Dougal, C. M., ‘Constructive homomorphisms for classical groups’, CIRCA Preprint 2009/16, http://www-circa.mcs.st-and.ac.uk/pre-prints.php, J. Symbol. Comput., to appear.Google Scholar
[17] Roney-Dougal, C. M., ‘Conjugacy of subgroups of the general linear group’, Experiment. Math. 13 (2004) 151–163.Google Scholar
[18] Rylands, L. J. and Taylor, D. E., ‘Matrix generators for the orthogonal groups’, J. Symbol. Comput. 25 (1998) 351–360.CrossRefGoogle Scholar
[19] Strassen, V., ‘Gaussian elimination is not optimal’, Numer. Math. 13 (1969) 354–356.CrossRefGoogle Scholar
[20] Taylor, D. E., The geometry of the classical groups (Heldermann, Berlin, 1992).Google Scholar
[21] Zassenhaus, H., ‘On the spinor norm’, Arch. Math. 13 (1962) 434–451.Google Scholar