Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T12:08:41.690Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructing Maximal Subgroups of Classical Groups

Published online by Cambridge University Press:  01 February 2010

Derek F. Holt  and
Colva M. Roney-Dougal
Derek F. Holt
Affiliation:
Mathematics Institute, The University of Warwick, Coventry, CV4 7AL, United Kingdom, [email protected], http://www.maths.warwick.ac.uk/~dfh/
Colva M. Roney-Dougal
Affiliation:
School of Computer Science, The University of St. Andrews, Fife KY16 9SS, United Kingdom, [email protected], http://www.dcs.st-and.ac.uk/~colva/

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.

The maximal subgroups of the finite classical groups are divided by a theorem of Aschbacher into nine classes. In this paper, the authors show how to construct those maximal subgroups of the finite classical groups of linear, symplectic or unitary type that lie in the first eight of these classes. The ninth class consists roughly of absolutely irreducible groups that are almost simple modulo scalars, other than classical groups over the same field in their natural representation. All of these constructions can be carried out in low-degree polynomial time.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 46 - 79
Copyright
Copyright © London Mathematical Society 2005

References

1.Aschbacher, M., ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984) 469–514.CrossRefGoogle Scholar
2.Beals, R., Leedham-Green, C. R., Niemeyer, A. C., Praeger, C. E. and Seress, Á., ‘A black-box group algorithm for recognizing finite symmetric and alternating groups. I’, Trans. Amer. Math. Soc. 355 (2003) 2097–2113.CrossRefGoogle Scholar
3.Bosma, W. and Cannon, J. J., eds, Handbook of MAGMA functions, edn 2.9 (School of Mathematics and Statistics, University of Sydney, 1995).Google Scholar
4.Bürgisser, P, Clausen, M. and Shokrollahi, M. A., Algebraic complexity theory (Springer, Berlin, 1990).Google Scholar
5.Cannon, J. J. and Holt, D. F., ‘Computing maximal subgroups of finite groups’, J. Symbolic Comput., to appear.Google Scholar
6.Carter, R. W., Simple groups of Lie type (Wiley, London/New York/Sydney/Toronto, 1972).Google Scholar
7.Dixon, J. D. and Mortimer, B., The primitive permutation groups of degree less than 1000’, Math. Proc. Cambridge Philos. Soc. 103 (1988) 213–238.CrossRefGoogle Scholar
8.Eick, B. and Hulpke, A., ‘Computing the maximal subgroups of a permutation group I’, Groups and computation III, Ohio, 1999 (ed. Kantor, W. M. and Seress, Á., Walter de Gruyter, Berlin/New York, 2001) 155–168.Google Scholar
9.Geddes, K. O., Czapor, S. R. and Labahn, G., Algorithms for computer algebra (Kluwer Academic Publishers, Boston/Dordrecht/London, 1992).CrossRefGoogle Scholar
10.Gorenstein, D., Finite groups (Harper and Row, New York/London, 1968).Google Scholar
11.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers, 5th edn (Oxford University Press, New York, 1979).Google Scholar
12.Hiss, G. and Malle, G., ‘Low dimensional representations of quasi-simple groups’, LMS J. Comput. Math. 4 (2001) 22–63; http://www.lms.ac.uk/jcm/4/lms2000-014.CrossRefGoogle Scholar
13.Holt, D. F. and Rees, S., ‘Testing modules for irreducibility’, J. Austral. Math. Soc. Ser. A 57 (1994) 1–16.Google Scholar
14.Kantor, W. M. and Seress, Á., ‘Black box classical groups’, Mem. Amer. Math. Soc. 149 (2001) no. 708, viii+168 pp.Google Scholar
15.Kleidman, P. and Liebeck, M. W., The subgroup structure of the finite classical groups (Cambridge Univ. Press, Cambridge, 1990).CrossRefGoogle Scholar
16.Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia Math. Appl. 20 (Addison-Wesley, Reading, MA, 1983).Google Scholar
17.Liebeck, M. W. and Seitz, G. M., ‘A survey of maximal subgroups of exceptional groups of Lie type’, Groups, combinatorics & geometry (Durham, 2001) (ed. Ivanov, A. A., Liebeck, M. W. and Saxl, J., World Scientific, River Edge, NJ, 2003) 139–146.CrossRefGoogle Scholar
18.Liebeck, M. W., Praeger, C. E. and Saxl, J., ‘A classification of the maximal subgroups of the finite alternating and symmetric groups’, J. Algebra 111 (1987) 365–383.CrossRefGoogle Scholar
19.Liebeck, M. W., Praeger, C. E. and Saxl, J., ‘The maximal factorizations of the finite simple groups and their automorphism groups’, Mem. Amer. Math. Soc. 86 (1990) no. 432, iv+151pp.Google Scholar
20.Lübeck, F., ‘Small degree representations of finite Chevalley groups in defining characteristic’, LMS J. Comput. Math. 4 (2001) 135–169; http://www.lms.ac.uk/jcm/4/lms2000-015.CrossRefGoogle Scholar
21.Roney-Dougal, C. M., ‘Conjugacy of subgroups of the general linear group’, Exp. Math. 13 (2004) 151–163.CrossRefGoogle Scholar
22.Strassen, V., ‘Gaussian elimination is not optimal’, Numer. Math. 13 (1969) 354–356.CrossRefGoogle Scholar
23.Taylor, D. E., ‘Pairs of generators for matrix groups, I’, The Cayley Bulletin 3 (1987) 76–85.Google Scholar
24.Wilson, R. A., Walsh, P., Tripp, J., Sulieiman, I., Rogers, S., Parker, R., Norton, S., Nickerson, S., Linton, S. and Bray, J., ATLAS of finite group representations, http://web.mat.bham.ac.uk/atlas/v2.0/.Google Scholar