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Generalised Sifting in Black-Box Groups

Published online by Cambridge University Press: 01 February 2010

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
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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
Copyright
Copyright © London Mathematical Society 2005

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