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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, [email protected], 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, [email protected], 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, [email protected], 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, [email protected], 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

References

1. Ahmad, ABD Ghafur Bin, ‘The unsolvability of efficiency for groups’, Southeast Asian Math. Bull. 22 (1998) 331–336.Google Scholar
2. Bosma, Wieb, Cannon, John and Playoust, Catherine, ‘The Magma algebra system, I: the user language’, J. Symbolic Comput. 24 (1997) 235–265; see also http://magma.maths.usyd.edu.au/magma/CrossRefGoogle Scholar
3. Campbell, C. M., and Robertson, E. F., ‘A deficiency zero presentation for SL (2, p)’, Bull. London Math. Soc. 12 (1980) 17–20.CrossRefGoogle Scholar
4. Campbell, C. M., Robertson, E. F., Kawamata, T., Miyamoto, I. and Williams, P.D., ‘Deficiency zero presentations for certain perfect groups’, Proc. Roy. Soc. Edinburgh 103A (1986) 63–71.Google Scholar
5. Campbell, Colin M. and Robertson, Edmund F., ‘The efficiency of simple groups of order < 105’, Comm. Algebra 10 (1982) 217–225.CrossRefGoogle Scholar
6. Campbell, Colin M. and Robertson, Edmund F., ‘On a class of groups related to SL(2, 2n)’, Computational group theory (ed. Atkinson, Michael D., Academic Press, London, 1984) 43–49.Google Scholar
7. Cambpell, Colin M., Havas, George, Hulpke, Alexander and Robertson, Edmund F., ‘Efficient simple groups’, Comm. Algebra 31 (2003) 5191–5197.CrossRefGoogle Scholar
8. Cambpell, Colin M., Robertson, E. F. and Williams, P. D., ‘Efficient presentations for finite simple groups and related groups’, Groups - Korea 1988, Lecture Notes in Math. 1398 (ed. Kim, A. C. and Neumann, B. H., Springer, New York, 1989) 65–72.Google Scholar
9. Cannon, John. J., McKay, John and Young, Kiang Chuen, ‘The non-abelian simple groups G, \G\ < 105 -presentations’, Comm. Algebra 7 (1979) 1397–1406.CrossRefGoogle Scholar
10. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Oxford Univ. Press, Oxford, 1985).Google Scholar
11. Coxeter, H. S. M., ‘The abstract groups Gmnp‘ Trans. Amer. Math. Soc. 45 (1939)73–150.Google Scholar
12. Doković, Dragomir Ž., ‘Presentations of some finite simple groups’, J. Austral. Math. Soc. Ser. A 45 (1988) 143–168.CrossRefGoogle Scholar
13. Gamble, Greg, Hulpke, Alexander, Havas, George and Ramsay, Colin, G AP package ACE; Advanced Coset Enumerator, (2004); http://www.gap-system.org/Packages/ace.html.Google Scholar
14. THE GAP GROUP, GAP - Groups, algorithms, and programming, Version 4.3, (2000;) http://www.gap-system.org/.Google Scholar
15. THE GAP GROUP, Balanced presentations for covering groups of simple groups, (2004); http://www.gap-system.org/Doc/Examples/balanced.html.Google Scholar
16. Havas, George and Ramsay, Colin, ‘Proving a group trivial made easy: a case study in coset enumeration’, Bull. Austral. Math. Soc. 62 (2000) 105–118.CrossRefGoogle Scholar
17. Havas, George and Ramsay, Colin, Coset enumeration: ACE version 3.001, (2001); http://www.itee.uq.edu.au/~havas/ace3001.tar.gz.CrossRefGoogle Scholar
18. Havas, George and Ramsay, Colin, ‘Short balanced presentations of perfect groups’, Groups St Andrews 2001 in Oxford, vol. 1, London Math. Soc. Lecture Note Ser. 304 (ed. Campbell, C. M., Robertson, E. F. and Smith, G. C., Cambridge Univ. Press, Cambridge, (2003) 238–243.Google Scholar
19. Havas, George, Newman, M. F. and O'Brien, E. A., ‘On the efficiency of some finite groups’, Comm. Algebra 32 (2004) 649–656.CrossRefGoogle Scholar
20. Kenne, P. E., ‘Efficient presentations for three simple groups’, Comm. Algebra 14 (1986) 797–800.CrossRefGoogle Scholar
21. The New York Group Theory Cooperative, Magnus; http://www.grouptheory.org/.Google Scholar
22. Robertson, Edmund. F., ‘Efficiency of finite simple groups and their covering groups’, Contemp. Math. 45 (1985) 287–294.CrossRefGoogle Scholar
23. Sunday, J. G., ‘Presentations of the groups SL(2, m) and PSL(2, m)’, Canad. J. Math. 24(1972)1129–1131.CrossRefGoogle Scholar
24. Wiegold, J., ‘The Schur multiplier: an elementary approach’, Groups - St Andrews 1981, London Math. Soc. Lecture Note Ser. 71 (ed. Campbell, C. M. and Robertson, E. F., Cambridge Univ. Press, Cambridge, 1982) 137–154.Google Scholar