Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-29T19:51:33.527Z Has data issue: false hasContentIssue false
  • English
  • Français

THE $(2,3)$ -GENERATION OF THE SPECIAL LINEAR GROUPS OVER FINITE FIELDS

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

Published online by Cambridge University Press:  26 September 2016

MARCO ANTONIO PELLEGRINI
MARCO ANTONIO PELLEGRINI*
Affiliation:
Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via Musei 41, 25121 Brescia, Italy email [email protected]
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.

We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$ -generated, that is, which are generated by an involution and an element of order $3$ . This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$ -generated.

Keywords

generationlinear 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 95 , Issue 1 , February 2017 , pp. 48 - 53
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Aschbacher, M. and Guralnick, R., ‘Some applications of the first cohomology group’, J. Algebra 90 (1984), 446–460.Google Scholar
Di Martino, L. and Vavilov, N., ‘(2, 3)-generation of SL(n, q). I. Cases n = 5, 6, 7’, Comm. Algebra 22 (1994), 1321–1347.CrossRefGoogle Scholar
Di Martino, L. and Vavilov, N., ‘(2, 3)-generation of SL(n, q). II. Cases n ≥ 8’, Comm. Algebra 24 (1996), 487–515.Google Scholar
Gencheva, E. and Genchev, Ts., ‘(2, 3)-generation of the special linear groups of dimension 8’, in: Mathematics and Education in Mathematics, 2015, Proc. Forty Fourth Spring Conf. Union of Bulgarian Mathematicians, SOK ‘Kamchia’, 2–6 April 2015 167–173.Google Scholar
Gencheva, E., Genchev, Ts. and Tabakov, K., ‘ $(2,3)$ -generation of the special linear groups of dimensions 9, 10 and 11’. arxiv.org/pdf/1412.8631v5.pdf.Google Scholar
King, C. S. H., ‘Generation of finite simple groups by an involution and an element of prime order’. arxiv.org/pdf/1603.04717v1.pdf.Google Scholar
Liebeck, M., Praeger, C. E. and Saxl, J., ‘Transitive subgroups of primitive permutation groups’, J. Algebra 234 (2000), 291–361.Google 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.Google Scholar
Lübeck, F. and Malle, G., ‘(2, 3)-generation of exceptional groups’, J. Lond. Math. Soc. (2) 59 (1999), 109–122.CrossRefGoogle Scholar
Macbeath, A. M., ‘Generators for the linear fractional groups’, Proc. Sympos. Pure Math. 12 (1969), 14–32.CrossRefGoogle Scholar
Miller, G. A., ‘On the groups generated by two operators’, Bull. Amer. Math. Soc. 7 (1901), 424–426.CrossRefGoogle Scholar
Pellegrini, M. A., ‘The (2, 3)-generation of the classical simple groups of dimensions 6 and 7’, Bull. Aust. Math. Soc. 93 (2016), 61–72.Google Scholar
Pellegrini, M. A., Prandelli, M. and Tamburini Bellani, M. C., ‘The (2, 3)-generation of the special unitary groups of dimension 6’, J. Algebra Appl. 15 1650171 (2016), 12 pages.CrossRefGoogle Scholar
Pellegrini, M. A. and Tamburini, M. C., ‘Finite simple groups of low rank: Hurwitz generation and (2, 3)-generation’, Int. J. Group Theory 4 (2015), 13–19.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. 14 1550148 (2015), 15 pages.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.Google Scholar
Steinberg, R., ‘Generators for simple groups’, Canad. J. Math. 14 (1962), 277–283.Google Scholar
Suzuki, M., ‘On a class of doubly transitive groups’, Ann. of Math. (2) 75 (1962), 105–145.Google Scholar
Tabakov, K., ‘(2, 3)-generation of the groups PSL7(q)’, in: Proc. Forty Second Spring Conf. Union of Bulgarian Mathematicians, Borovetz, 2–6 April 2013 260–264.Google Scholar
Tabakov, K. and Tchakerian, K., ‘(2, 3)-generation of the groups PSL6(q)’, Serdica Math. J. 37 (2011), 365–370.Google Scholar
Tamburini, M. C., ‘Generation of certain simple groups by elements of small order’, Istit. Lombardo Accad. Sci. Lett. Rend. A 121 (1987), 21–27.Google Scholar
Tamburini, M. C., ‘The (2, 3)-generation of matrix groups over the integers’, in: Ischia Group Theory 2008 (World Scientific, Hackensack, NJ, 2009), 258–264.Google Scholar
Tamburini, M. C. and Wilson, J. S., ‘On the (2, 3)-generation of some classical groups. II’, J. Algebra 176 (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 (1994), 353–370.CrossRefGoogle Scholar
Tamburini Bellani, M. C. and Vsemirnov, M., ‘Hurwitz generation of PSp6(q)’, Comm. Algebra 43 (2015), 4159–4169.Google Scholar
Vsemirnov, M. A., ‘Is the group SL(6, Z) (2, 3)-generated?’, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 330 (2006), Vopr. Teor. Predst. Algebr. i Grupp. 13, 101–130, 272; English translation in J. Math. Sci. (N. Y.) 140 (2007), 660–675.Google Scholar
Vsemirnov, M. A., ‘On the (2, 3)-generation of matrix groups over the ring of integers’, Algebra i Analiz 19 (2007), 22–58; English translation in St. Petersburg Math. J. 19 (2008), 883–910.Google Scholar
Vsemirnov, M. A., ‘On the (2, 3)-generation of small rank matrix groups over integers’, Quad. Semin. Mat. Brescia 30 (2008), 1–15.Google Scholar
Woldar, A. J., ‘On Hurwitz generation and genus actions of sporadic groups’, Illinois Math. J. (3) 33 (1989), 416–437.Google Scholar