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Generalised Sifting in Black-Box Groups

Published online by Cambridge University Press:  01 February 2010

Sophie Ambrose ,
Max Neunhöffer ,
Cheryl E. Praeger  and
Csaba Schneider
Sophie Ambrose
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway Crawley, Western Australia 6009, Australia, sambrose@maths.uwa.edu.au, http://www.maths.uwa.edu.au/~sambrose
Max Neunhöffer
Affiliation:
Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule Aachen, Templergraben 64, 52056 Aachen, Germany, max.neunhoeffer@math.rwth-aachen.de, http://www.math.rwth-aachen.de/~Max.Neunhoeffer
Cheryl E. Praeger
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway Crawley, Western Australia 6009, Australia, praeger@maths.uwa.edu.au, http://www.maths.uwa.edu.au/~praeger
Csaba Schneider
Affiliation:
Informatics Laboratory, Computer and Automation Research Institute, The Hungarian Academy of Sciences, 1518 Budapest Pf. 63, Hungary, csaba.schneider@Sztaki.hu, http://www.sztaki.hu/~schneider

Abstract

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This paper presents a generalisation of the sifting procedure introduced originally by Sims for computation with finite permutation groups, and now used for many computational procedures for groups, such as membership testing and finding group orders. The new procedure is a Monte Carlo algorithm, and it is presented and analysed in the context of black-box groups. It is based on a chain of subsets instead of a subgroup chain. Two general versions of the procedure are worked out in detail, and applications are given for membership tests for several of the sporadic simple groups. The authors' major objective was that the procedures could be proved to be Monte Carlo algorithms, and the costs computed. In addition, they explicitly determined suitable subset chains for six of the sporadic groups, and then implemented the algorithms involving these chains in the GAP computational algebra system. It turns out that sample imple-mentations perform well in practice. The implementations will be made available publicly in the form of a GAP package.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 217 - 250
Copyright
Copyright © London Mathematical Society 2005

References

1.Beals, Robert M., Leedham-Green, Charles R., Niemeyer, Alice C., Praeger, Cheryl E. and Seress, Ákos, ‘Permutations with restricted cycle structure and an algorithmic application’, Combin. Probab. Comput. 11 (2002) 447–464.CrossRefGoogle Scholar
2.Beals, Robert M., Leedham-Green, Charles R., Niemeyer, Alice C., Praeger, Cheryl E. and Seress, Ákos, ‘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, Wieb, Cannon, John and Playoust, Catherine, ‘The Magma algebra system I: The user language’, J.Symbolic Comput. 24 (1997) 235–265.CrossRefGoogle Scholar
4.Bratus, Sergey and Pak, Igor, ‘Fast constructive recognition of a black box group isomorphic to Sn or An using Goldbach's conjecture’, J.Symbolic Comput. 29 (2000) 33–57.CrossRefGoogle Scholar
5.Celler, F. and Leedham-Green, C.R.‘A constructive recognition algorithm for the special linear group’, The Atlas of finite groups: ten years on{Birmingham, 1995) (Cambridge Univ.Press, Cambridge, 1998) 11–26.Google Scholar
6.Cooperman, Gene, Finkelstein, Larry and Linton, Steve, ‘Constructive recognition of a black box group isomorphic to GL(n, 2)’, Groups and computation II (NewBrunswick, NJ, 1995)(Amer. Math. Soc, Providence, RI, 1997) 85–100.Google Scholar
7.Conway, J.H., Curtis, R.T., Norton, S.P., Parker, R.A. and Wilson, R.A., Atlas of finite groups, (Oxford University Press, Eynsham, 1985).Google Scholar
8. The Gap ‘GAP - groups, algorithms, and programming’, Version 4.4, 2004, http://www.gap-system.org.Google Scholar
9.Holmes, P.E., Linton, S.A., O'Brien, E.A., Ryba, A.J.E. and Wilson, R.A.,‘Constructive membership testing in black-box groups’, preprint, Queen Mary, University of London, 2004.Google Scholar
10.Kantor, William M. and Seress, Ákos, ‘Black box classical groups’, Mem. Amer. Math.Soc. 149 (708) (2001).Google Scholar
11.Leedham-Green, Charles R., ‘The computational matrix group project’, Groups and computation III, OSU Math.Res.Inst.Publ.(ed.Kantor, William M. and Seress, Ákos, Walter de Gruyter, 2000) 85–101.Google Scholar
12.Leedham-Green, Charles R., Niemeyer, Alice C., O'Brien, E.A. and Praeger, Cheryl E., ‘Recognising matrix groups over finite fields’, Computer algebra handbook. Foundations, applications, systems (ed. Weispfenning, V., Grabmeier, J. and Kaltofen, E., Springer, Berlin/New York, 2003) 459–460.Google Scholar
13.Robinson, Derek J.S., A course in the theory of groups (Springer, New York/ Heidelberg/Berlin, 1982).Google Scholar
14.Seress, Ákos, Permutation group algorithms(Cambridge Univ. Press, 2003).CrossRefGoogle Scholar
15.Sims, Charles C., ‘Computational methods in the study of permutation groups’, Computational problems in abstract algebra, Proc.Conf.Oxford, 1967 (Pergamon Press, Oxford, 1970) 169–183.Google Scholar
16.Sims, Charles C., ‘Computing with subgroups of automorphism groups of finite groups’, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, (Kihei, HI) (ACM, New York, 1997) 400–403(electronic).CrossRefGoogle Scholar
17.Wilson, R.A., ‘Standard generators for sporadic simple groups’, J.Algebra. 184 (1996) 505–515.CrossRefGoogle Scholar
18.Wilson, R.A. et al. , ‘ATLAS of finite group representations’, 2004, http://brauer.maths.qmul.ac.uk/Atlas/.Google Scholar