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A recognition algorithm for non-generic classical groups over finite fields

Part of: Linear algebraic groups and related topics Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  09 April 2009

Azice C. Niemeyer  and
Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
Department of Mathematics and Statistics Univesity of Western Australia Nedlands WA 6907 Australia
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Abstract

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In a previous paper the authors described an algorithm to determine whether a group of matrices over a finite field, generated by a given set of matrices, contains one of the classical groups or the special linear group. The algorithm was designed to work for all sufficiently large field sizes and dimensions of the matrix group. However, it did not apply to certain small cases. Here we present an algorithm to handle the remaining cases. The theoretical background of the algorithm presented in this paper is a substantial extension of that needed for the original algorithm.

Keywords

matrix groupsrecognition algorithmnon-generic.

MSC classification

Secondary: 20G40: Linear algebraic groups over finite fields 60B99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 67 , Issue 2 , October 1999 , pp. 223 - 253
Copyright
Copyright © Australian Mathematical Society 1999

References

[1]Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
[2]Dickson, L. E., Linear groups with an exposition of the Galois field theory, with an introduction by Magnus, W. (Dover Publications Inc., New York, 1958).Google Scholar
[3]Fiet, W., ‘On large Zsigmondy primes’, Proc. Amer. Math. Soc. 102 (1988), 29–36.Google Scholar
[4]Guralnick, R. M., Penttila, T., Praeger, C. E. and Saxl, J., ‘Linear groups with orders having certain primitive prime divisors’, Proc. London Math. Soc. 78 (1999), 167–214.Google Scholar
[5]Hering, C., ‘Transitive linear groups and linear groups which contain irreducible subgroups of prime order’, Geom. Dedicata 2 (1974), 425–460.Google Scholar
[6]Holt, Derek F. and Rees, Sarah, ‘Testing modules for irreducibility’, J. Austral. Math. Soc. Ser. A 57 (1994), 1–16.Google Scholar
[7]Kleidman, P. and Liebeck, M., The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser. vol. 129 (Cambridge University Press, Cambridge, 1990).Google Scholar
[8]Neumann, P. M. and Praeger, C. E., ‘A recognition algorithm for special linear groups’, Proc. London Math. Soc. 65 (1992), 555–603.Google Scholar
[9]Niemeyer, A. C. and Praeger, C. E., ‘Implementing a recognition algorithm for classical groups’, in: Groups and computation II (eds. Finkelstein, L. and Kantor, W. M.), Amer. Math. Soc. DIMACS Series vol. 28 (DIMACS, 1995) (1997) pp. 273–296.Google Scholar
[10]Niemeyer, A. C. and Praeger, C. E., ‘A recognition algorithum for classical groups over finite fields’, Proc. London Math. Soc. (3) 77 (1998), 117–169.CrossRefGoogle Scholar
[11]Parker, R. A., ‘The computer calculation of modular characters (the Mear-Axe)’, in: Computaional group theory (ed. Atkinson, M. D.), (Durham, 1982) (Academic Press, New York, 1984) pp. 267–274.Google Scholar
[12]Taylor, D. E., The geometry of the classical groups, Signa Series in Pure Mathematics vol. 9 (Heldermann, Berlin, 1992).Google Scholar
[13]Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. Phys. 3 (1892), 265–284.Google Scholar