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Nice Efficient Presentions for all Small Simple Groups and their Covers

Published online by Cambridge University Press:  01 February 2010

Colin M. Campbell ,
George Havas ,
Colin Ramsay  and
Edmund F. Robertson
Colin M. Campbell
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom, colinc@mcs.st-and.ac.uk, http://www-groups.mcs.st-andrews.ac.uk/~colinc/
George Havas
Affiliation:
ARC Centre for Complex Systems, School of Information Technology and Electrical Engineering, The University of Queensland, Queensland 4072, Australiahavas@itee.uq.edu.au, http://www.itee.uq.edu.au/~havas/
Colin Ramsay
Affiliation:
ARC Centre for Complex Systems, School of Information Technology and Electrical Engineering, The University of Queensland, Queensland 4072, Australia, cram@itee.uq.edu.au, http://www.itee.uq.edu.au/~cram/
Edmund F. Robertson
Affiliation:
School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom, edmund@mcs.st-and.ac.uk, http://www-groups.mcs.st-andrews.ac.uk/~edmund/

Abstract

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Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors‘presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in partic-ular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 7 , 2004 , pp. 266 - 283
Copyright
Copyright © London Mathematical Society 2004

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