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    Guralnick, R.M. Kantor, W.M. Kassabov, M. and Lubotzky, A. 2011. Remarks on proficient groups. Journal of Algebra, Vol. 326, Issue. 1, p. 169.

  • LMS Journal of Computation and Mathematics, Volume 7
  • January 2004, pp. 266-283

Nice Efficient Presentions for all Small Simple Groups and their Covers

  • Colin M. Campbell (a1), George Havas (a2), Colin Ramsay (a3) and Edmund F. Robertson (a4)
  • DOI:
  • Published online: 01 February 2010

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.

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5.Colin M. Campbell and Edmund F. Robertson , ‘The efficiency of simple groups of order < 105, Comm. Algebra 10 (1982) 217225.

7.Colin M. Cambpell , George Havas , Alexander Hulpke and Edmund F. Robertson , ‘Efficient simple groups’, Comm. Algebra 31 (2003) 51915197.

9.John. J. Cannon , John McKay and Kiang Chuen Young , ‘The non-abelian simple groups G, \G\ < 105 -presentations’, Comm. Algebra 7 (1979) 13971406.

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19.George Havas , M. F. Newman and E. A. O'Brien , ‘On the efficiency of some finite groups’, Comm. Algebra 32 (2004) 649656.

20.P. E. Kenne , ‘Efficient presentations for three simple groups’, Comm. Algebra 14 (1986) 797800.

21. The New York Group Theory Cooperative, Magnus;

22.Edmund. F. Robertson , ‘Efficiency of finite simple groups and their covering groups’, Contemp. Math. 45 (1985) 287294.

23.J. G. Sunday , ‘Presentations of the groups SL(2, m) and PSL(2, m)’, Canad. J. Math. 24(1972)11291131.

24.J. Wiegold , ‘The Schur multiplier: an elementary approach’, Groups - St Andrews 1981, London Math. Soc. Lecture Note Ser. 71 (ed. C. M. Campbell and E. F. Robertson , Cambridge Univ. Press, Cambridge, 1982) 137154.

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LMS Journal of Computation and Mathematics
  • ISSN: -
  • EISSN: 1461-1570
  • URL: /core/journals/lms-journal-of-computation-and-mathematics
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