All simple groups with order from 1 million to 5 million are efficient

Document Type: Research Paper

Authors

1 School of Mathematics and Statistics, University of St Andrews

2 Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland

3 Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland

Abstract

‎There is much interest in finding short presentations for the finite‎ ‎simple groups‎. ‎Indeed it has been suggested that all these groups are‎ ‎efficient in a technical sense‎. ‎In previous papers we produced nice‎ ‎efficient presentations for all except one of the simple groups with‎ ‎order less than one million‎. ‎Here we show that all simple groups with‎ ‎order between $1$ million and $5$ million are efficient by giving efficient‎ ‎presentations for all of them‎. ‎Apart from some linear groups these‎ ‎results are all new‎. ‎We also show that some covering groups and‎ ‎some larger simple groups are efficient‎. ‎We make substantial use of‎ ‎systems for computational group theory and‎, ‎in particular‎, ‎of computer‎ ‎implementations of coset enumeration to find and verify our presentations‎.

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W. Bosma, J. Cannon and C. Playoust (1997). The Magma algebra system I: the user language. J. Symbolic Comput., See also http://magma.maths.usyd.edu.au/magma/.. 24, 235-265
J. N. Bray, M. D. E. Conder, C. R. Leedham-Green and E. A. O'Brien (2011). Short presentations for alternating and symmetric groups. Trans. Amer. Math. Soc.. 363, 3277-3285
C. M. Campbell, G. Havas, C. Ramsay and E. F. Robertson (2004). Nice efficient presentations for all small simple groups and their covers. LMS J. Comput. Math.. 7, 266-283
C. M. Campbell, G. Havas, C. Ramsay and E. F. Robertson (2007). On the efficiency of the simple groups with order less than a million and their covers. Experiment. Math.. 16, 347-358
C. M. Campbell and E. F. Robertson (1980). A deficiency zero presentation for SL(2,p). Bull. London Math. Soc.. 12, 17-20
C. M. Campbell and E. F. Robertson (1988). Computing with finite simple groups and their covering groups. Computers in Algebra, ed. M. C. Tangora and M. Dekker, New York. , 17-26
C. M. Campbell, E. F. Robertson and P. D. Williams (1989). Efficient presentations for finite simple groups and related groups. Groups-Korea 1988, Lecture Notes in Mathematics, 1398, Springer, Berlin. , 65-72
M. Conder, G. Havas and M. F. Newman (2011). On one-relator quotients of the modular group. Groups St Andrews 2009 in Bath, Volume I, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge. 387, 183-197
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson (1985). Atlas of finite groups. Oxford University Press, Oxford.
D. B. A. Epstein (1961). Finite presentations of groups and 3-manifolds. Quart. J. Math. Oxford Ser. (2). 12, 205-212
The GAP~Group (2012). GAP -- Groups, Algorithms, and Programming -- a System for Computational Discrete Algebra. http://www.gap-system.org/.
R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky (2008). Presentations of finite simple groups: a quantitative approach. J. Amer. Math. Soc.. 21, 711-774
R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky (2011). Presentations of finite simple groups: a computational approach. J. Eur. Math. Soc.. 13, 391-458
G. Havas and D. F. Holt (2010). On Coxeter's families of group presentations. J. Algebra. 324, 1076-1082
G. Havas and C. Ramsay (2001). Coset enumeration: ACE version 3.001. http://www.itee.uq.edu.au/~havas/ace3001.tar.gz.
G. Havas and C. Ramsay (2003). Short balanced presentations of perfect groups. Groups St Andrews 2001 in Oxford, Volume 1, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge. 304, 238-243
G. Havas and C. Ramsay (2003). Breadth-first search and the Andrews-Curtis conjecture. Internat. J. Algebra Comput.. 13, 61-68
I. Korchagina and A. Lubotzky (2006). On presentations and second cohomology of some finite simple groups. Publ. Math. Debrecen. 69, 341-352
J. G.~Sunday (1972). Presentations of the groups SL(2,m) and PSL(2,m). Canad. J. Math.. 24, 1129-1131
J. S. Wilson (2006). Finite axiomatization of finite soluble groups. J. London Math. Soc. (2). 74, 566-582
R. Wilson, P. Walsh, J. Tripp, Ibrahim Suleiman, Richard Parker, Simon Norton, Simon Nickerson, Steve Linton, John Bray and Rachel Abbott (2012). ATLAS of Finite Group Representations - Version 3. http://brauer.maths.qmul.ac.uk/Atlas/v3/.