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Fast Constructive Recognition of Black-Box Unitary Groups

Published online by Cambridge University Press:  01 February 2010

Peter A. Brooksbank
Peter A. Brooksbank
Affiliation:
Department of Mathematics, The Ohio State University, 231 W. 18th Ave., Columbus OH 43210, USAbrooksbank@math.ohio-state.edu, http://www.math.ohio-state.edu/~brooksbank

Abstract

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In this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 6 , 2003 , pp. 162 - 197
Copyright
Copyright © London Mathematical Society 2003

References

1Aschbacher, M. ‘On the maximal subgroups of the finite classical groups’, Invent. Math. 76 (1984) 469–514.CrossRefGoogle Scholar
2Babai, L., ‘Local expansion of vertex-transitive graphs and random generation in finite groups’, Proc. ACM Symp. on Theory of Computing (1991) 164–174.Google Scholar
3Babai, L. and Beals, R., ‘A polynomial-time theory of black box groups I’, Groups St Andrews 1997 in Bath, I, London Math. Soc. Lecture Note Ser. 260 (ed. Campbell, C.M., Robertson, E.F., Ruskuc, N., and Smith, G.C., Cambridge University Press, 1999) 30–64.CrossRefGoogle Scholar
4Beals, R. and Babai, L., ‘Las Vegas algorithms for matrix groups’, Proc. IEEE Symp. Found. Comp. Sci. (1993) 427–36. 162Google Scholar
5Beals, R., Leedham-Green, C.R., Niemeyer, A.C., Praeger, C.E. and Seress, Á., ‘A black box algorithm for recognising finite symmetric and alternating groups, I‘, Trans. Amer. Math, Soc. 355 (2003) 2097–2113 (electronic).CrossRefGoogle Scholar
6Brooksbank, P.A., ‘Constructive recognition of classical groups in their natural representation’, J. Symbolic Comput. 35 (2003) 195–239.CrossRefGoogle Scholar
7Brooksbank, P.A., ‘Constructive recognition of the finite simple classical groups’, Ph. D. thesis, Univeristy of Oregon, 2001.Google Scholar
8Brooksbank, P. A. and Kantor, W.M., ‘On constructive recognition of a black box PSL(d, q)’, Groups and Computation III, Ohio State Univ. Math. Res. Inst. Publ. 8, (ed. Kantor, W.M. and Seress, Á., de Gruyter, Walter, Berlin/New York, 2001) 95–111.CrossRefGoogle Scholar
9Conder, M. and Leedham-Green, C.R., ‘Fast recognition of classical groups over large fields’, Groups and computation III, Ohio State Univ. Math. Res. Inst. Publ. 8, (ed. Kantor, W.M. and Seress, Á., de Gruyter, Walter, Berlin/New York, 2001) 113–121.CrossRefGoogle Scholar
10Cooperman, G., Finkelstein, L. and Linton, S., ‘Constructive recognition of a black box group isomorphic to GL(n, 2)’, Groups and computation II, Proceedings of a DI-MACS Workshop (ed. Finkelstein, L. and Kantor, W.M., Amer. Math. Soc, Providence, RI, 1997) 85–100.CrossRefGoogle Scholar
11The GAP Group, ‘Groups, algorithms, and programming’, Version 4.2 (The GAP Group, Aachen, St Andrews); http://www-gap.dcs.st-and.ac.uk/gap.Google Scholar
12Kantor, W.M. and Liebler, R.A., ‘The rank 3 permutation representations of the finite classical groups’, Trans. Amer. Math. Soc. 271 (1982) 1–71.CrossRefGoogle Scholar
13Kantor, W.M. and Seress, Á., ‘Black box classical groups’, Mem. Amer. Math. Soc. 149 (2001).Google Scholar
14Kantor, W.M. and Seress, Á., ‘Computing with matrix groups’, Groups, combinatorics and geometry (Durham 2001) (ed. Ivanov, A.A., Liebeck, M.W and Saxl, J, World Scientific, 2003).Google Scholar
15Kleidman, P.B. and Liebeck, M.W, The subgroup structure of the finite classical groups, London Math. Soc. Lecture Note Ser. 129 (Cambridge Univ. Press, 1990).CrossRefGoogle Scholar
16Leedham-Green, C.R., ‘The Computational Matrix Group Project’, Groups and computation III, Ohio State Univ. Math. Res. Inst. Publ. 8 (ed. Kantor, W.M. and Seress, Á, Gruyter, Walter de, Berlin/New York, 2001) 229–247.CrossRefGoogle Scholar
17Neumann, P.M. and Praeger, C.E., ‘A recognition algorithm for special linear groups–, Proc. London Math. Soc. (3) 65 (1992) 555–603.CrossRefGoogle Scholar
18Niemeyer, A.C. and Praeger, C.E., ‘A recognition algorithm for classical groups over finite fields’, Proc. London Math. Soc. (1) 77 (1998) 117–169.CrossRefGoogle Scholar
19Seress, À., Permutation group algorithms, Cambridge Tracts in Math. 152 (Cambridge University Press, 2003).CrossRefGoogle Scholar
20Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. Phys. 3 (1892) 265–284.CrossRefGoogle Scholar