LMS Journal of Computation and Mathematics

Research Article

Nice Efficient Presentions for all Small Simple Groups and their Covers

Colin M. Campbella1, George Havasa2, Colin Ramsaya3 and Edmund F. Robertsona4

a1 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/

a2 ARC Centre for Complex Systems, School of Information Technology and Electrical Engineering, The University of Queensland, Queensland 4072, Australia havas@itee.uq.edu.au, http://www.itee.uq.edu.au/~havas/

a3 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/

a4 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

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.

(Received October 03 2003)

(Revised September 16 2004)

(Accepted October 18 2004)