Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T21:23:59.310Z Has data issue: false hasContentIssue false
  • English
  • Français

Improved Bounds for the Spread of Sporadic Groups

Published online by Cambridge University Press:  01 February 2010

J. D. Bradley  and
P. E. Holmes
J. D. Bradley
Affiliation:
National University of Rwanda, Butare, Rwanda, johnbradleyl20@gmail.com
P. E. Holmes
Affiliation:
DPMMS, Centre for Mathematical Sciences, Cambridge CB3 OBW, United Kingdom, peh27@dpmms.cam.ac.uk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The spread of a group G is the greatest number r such that, for every set of non-trivial elements {x1,…,xr}, there exists an element y with the property that 〈xi, y〉 = G for 1 ≤ i ≤ r. In this paper we obtain good upper bounds for the spread of fourteen sporadic simple groups computationally, and we determine the value of the spread of M11 by hand.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007 , pp. 132 - 140
Copyright
Copyright © London Mathematical Society 2007

References

1.Binder, G., ‘The bases of the symmetric group’, Izv. Vyss. Ucebn. Zaved. Mathematika 78 (1968) 19–25.Google Scholar
2.Binder, G., ‘The two element bases of the symmetric group’, Izv. Vyss. Ucebn. Zaved. Mathematika 90 (1970) 9–11.Google Scholar
3.Blackburn, S., ‘Sets of permutations that generate the symmetric group pairwise’, J. Combin. Theory Ser. A 113 (2006) 1132–1138.CrossRefGoogle Scholar
4.Bosma, W. and Cannon, J. J., Handbook of Magma functions (School of Mathematics and Statistics, University of Sydney, Sydney, 1995).Google Scholar
5.Bradley, J. D. and Moori, J., ‘On the exact spread of sporadic simple groups’, Comm. Algebra, to appear.Google Scholar
6.Brenner, J. L. and Wiegold, J., ‘Two generator groups, I’, Michigan Math. J. 22 (1975) 53–64.CrossRefGoogle Scholar
7.Breuer, T., Guralnick, R. M. and Kantor, W. M.. ‘Probabilistic generation of finite simple groups II’, J. Algebra, to appear.Google Scholar
8.Conway, J. H.. Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Clarendon Press, 1985).Google Scholar
9.Conway, J. H. and Sloane, N. A., Sphere packings lattices and groups. (Springer, 1988).CrossRefGoogle Scholar
10.Ganief, S. and Moori, J., ‘On the spread of sporadic simple groups’, Comm. Algebra 29 (2001) 3239–3255.Google Scholar
11.Guralnick, R. M. and Kantor, W. M., ‘Probabilistic generation of finite simple groups’, J. Algebra 234 (2000) 743–792.CrossRefGoogle Scholar
12.Guralnick, R. M. and Shalev, A., ‘On the spread of finite simple groups’, Combinatorica 23 (2003) 73–87.CrossRefGoogle Scholar
13.Holmes, P. E., ‘Subgroup coverings of some sporadic groups’, J. Combin. Theory, Ser. A, to appear.Google Scholar
14.Holmes, P. E. and Maróti, A., ‘Covering and generating sporadic simple group pairwise’, Preprint, CIRCA, St. Andrews, 7 (2006).Google Scholar
15.Kantor, W. M. and Lubotzky, A., ‘The probability of generating a finite classical group’, Geom. Dedicate 36 (1990) 67–87.Google Scholar
16.Mathieu, E., ‘Memoire sur l'etude des fonctions de plusieurs quantities’, J. Math. Pures Appl. 6 (1861) 241–243.Google Scholar
17.Mathieu, E., ‘Sur les fonctions cinq fois transitives de 24 quantites’, J. Math. Pures Appl. 18 (1873) 25–46.Google Scholar
18.Wilson, R. A. et al. , ‘A World-Wide-Web Atlas of group representations’, http://brauer.maths.qmul.ac.uk/Atlas/v3/.Google Scholar
19.Woldar, A., ‘The exact spread of the Mathieu group M11’. J. Group Theory, to appear.Google Scholar