Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T12:41:47.507Z Has data issue: false hasContentIssue false
  • English
  • Français

THE (2,3)-GENERATION OF THE CLASSICAL SIMPLE GROUPS OF DIMENSIONS 6 AND 7

Part of: Special aspects of infinite or finite groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  05 August 2015

MARCO ANTONIO PELLEGRINI
MARCO ANTONIO PELLEGRINI*
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, I-25121 Brescia, Italy email marcoantonio.pellegrini@unicatt.it
Rights & Permissions [Opens in a new window]

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.

In this paper, we prove that the finite simple groups $\text{PSp}_{6}(q)$, ${\rm\Omega}_{7}(q)$ and $\text{PSU}_{7}(q^{2})$ are $(2,3)$-generated for all $q$. In particular, this result completes the classification of the $(2,3)$-generated finite classical simple groups up to dimension 7.

Keywords

generationclassical groups

MSC classification

Primary: 20G40: Linear algebraic groups over finite fields
Secondary: 20F05: Generators, relations, and presentations
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 61 - 72
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Aschbacher, M., ‘Chevalley groups of type G 2 as the group of a trilinear form’, J. Algebra 109(1) (1987), 193–259.CrossRefGoogle Scholar
Aschbacher, M. and Guralnick, R., ‘Some applications of the first cohomology group’, J. Algebra 90 (1984), 446–460.CrossRefGoogle Scholar
Bray, J. N., Holt, D. F. and Roney-Dougal, C. M., The Maximal Subgroups of the Low-Dimensional Finite Classical Groups, London Mathematical Society Lecture Note Series, 407 (Cambridge University Press, Cambridge, 2013).CrossRefGoogle Scholar
Carter, R. W., Simple Groups of Lie Type, Pure and Applied Mathematics, 28 (John Wiley, London–New York–Sydney, 1972).Google Scholar
Di Martino, L. and Tamburini, M. C., ‘2-Generation of finite simple groups and some related topics’, in: Generators and Relations in Groups and Geometries (eds. Barlotti, A. et al. ) (Kluwer Academic, Dordrecht, 1991), 195–233.CrossRefGoogle Scholar
Guralnick, R. M. and Kantor, W. M., ‘Probabilistic generation of finite simple groups’, J. Algebra 234(2) (2000), 743–792.CrossRefGoogle Scholar
Liebeck, M. W. and Shalev, A., ‘Classical groups, probabilistic methods, and the (2, 3)-generation problem’, Ann. of Math. (2) 144 (1996), 77–125.CrossRefGoogle Scholar
Lübeck, F. and Malle, G., ‘(2, 3)-generation of exceptional groups’, J. Lond. Math. Soc. (2) 59(1) (1999), 109–122.CrossRefGoogle Scholar
Macbeath, A. M., ‘Generators for the linear fractional groups’, Proc. Sympos. Pure Math. 12 (1969), 14–32.CrossRefGoogle Scholar
Malle, G., Saxl, J. and Weigel, T., ‘Generation of classical groups’, Geom. Dedicata 49(1) (1994), 85–116.CrossRefGoogle Scholar
Mazurov, V. and Khukhro, E., Kourovka Notebook No. 18, Unsolved Problems in Group Theory, arXiv:1401.0300.Google Scholar
Miller, G. A., ‘On the groups generated by two operators’, Bull. Amer. Math. Soc. 7 (1901), 424–426.CrossRefGoogle Scholar
Pellegrini, M. A., Prandelli, M. and Tamburini Bellani, M. C., ‘The $(2,3)$-generation of the special unitary groups of dimension $6$’, arXiv:1409.3411.Google Scholar
Pellegrini, M. A. and Tamburini Bellani, M. C., ‘The simple classical groups of dimension less than $6$ which are $(2,3)$-generated’, J. Algebra Appl., to appear, doi:10.1142/S0219498815501480.CrossRefGoogle Scholar
Pellegrini, M. A. and Tamburini Bellani, M. C., ‘Scott’s formula and Hurwitz groups’, arXiv:1501.07495.Google Scholar
Pellegrini, M. A., Tamburini Bellani, M. C. and Vsemirnov, M. A., ‘Uniform (2, k)-generation of the 4-dimensional classical groups’, J. Algebra 369 (2012), 322–350.CrossRefGoogle Scholar
Sanchini, P. and Tamburini, M. C., ‘(2, 3)-generation: a permutational approach’, Rend. Sem. Mat. Fis. Milano 64 (1994), 141–158.CrossRefGoogle Scholar
Steinberg, R., ‘Generators for simple groups’, Canad. J. Math. 14 (1962), 277–283.CrossRefGoogle Scholar
Tabakov, K., ‘(2, 3)-generation of the groups PSL6(q)’, Serdica Math J. 37(4) (2011), 365–370.Google Scholar
Tabakov, K., $(2,3)$-generation of the groups $\text{PSL}_{7}(q)$, Proc. Forty Second Spring Conf. Union of Bulgarian Mathematicians, Borovetz, 2–6 April 2013.Google Scholar
Tamburini, M. C. and Vsemirnov, M., ‘Irreducible (2, 3, 7)-subgroups of PGLn(F), n ≤ 7’, J. Algebra 300 (2006), 339–362.CrossRefGoogle Scholar
Tamburini, M. C. and Wilson, J. S., ‘On the (2, 3)-generation of some classical groups II’, J. Algebra 176(2) (1995), 667–680.CrossRefGoogle Scholar
Tamburini, M. C., Wilson, J. S. and Gavioli, N., ‘On the (2, 3)-generation of some classical groups I’, J. Algebra 168(1) (1994), 353–370.CrossRefGoogle Scholar
Tamburini Bellani, M. C. and Vsemirnov, M., ‘Hurwitz generation of $\text{PSp}_{6}(q)$’, Comm. Algebra, to appear.Google Scholar
Taylor, D. E., The Geometry of the Classical Groups, Sigma Series in Pure Mathematics, 9 (Heldermann, Berlin, 1992).Google Scholar
Vsemirnov, M., ‘More classical groups which are not (2, 3)-generated’, Arch. Math. (Basel) 96(2) (2011), 123–129.CrossRefGoogle Scholar