Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-14T07:49:53.996Z Has data issue: false hasContentIssue false
  • English
  • Français

Non-Hurwitz Classical Groups

Published online by Cambridge University Press:  01 February 2010

R. Vincent and
A.E. Zalesski
R. Vincent
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR47TJ, United Kingdom, Robert.Vincent@uea.ac.uk
A.E. Zalesski
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR47TJ, United Kingdom, a.zalesskii@uea.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.

In previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra 28 (2000) 5383–5404] it is shown that certain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the special linear and special unitary groups in dimensions 8.9,11.13. We also show that all groups Sp(n, q) are not Hurwitz for q even and n = 6,8,12,16. In the range 11 < n < 32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3), PΩ±(8, 3k), PΩ±10k), Ω(11,3k), Ω±(14,3k), Ω±(16,7k), Ω(n, 7k) for n = 9,11,13, PSp(8, 7k) are not Hurwitz.

Information

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 10 , 2007, pp. 21 - 82
Copyright
Copyright © London Mathematical Society 2007

References

1. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., ATLAS of finite groups (Clarendon Press, Oxford, 1985).Google Scholar
2. Jansen, C., Lux, K., Parker, R. A. and Wilson, R. A., An Atlas of Brauer characters (Clarendon Press, Oxford. 1995).Google Scholar
3. Cohen, J. M., ‘On non-Hurwitz groups and non-congruence subgroups of the modular group’, Glasgow Math. J. 22 (1981) 1–7.CrossRefGoogle Scholar
4. Conder, M. D. E., ‘Generators for alternating and symmetric groups’, J. London Math. Soc. (2) 22 (1980) 75–86.10.1112/jlms/s2-22.1.75CrossRefGoogle Scholar
5. Conder, M. D. E., ‘Hurwitz groups: a brief survey’, Bull. Arner. Math. Soc. 23 (1990) 359–370.10.1090/S0273-0979-1990-15933-6CrossRefGoogle Scholar
6. Martino, L. Di, Tamburini, CH. and Zalesski, A., ‘Hurwitz groups of small rank’, Comm. Algebra 28 (2000) 5383–5404.CrossRefGoogle Scholar
7. Feit, W., The representation theory of finite groups (North-Holland, Ams terdam, 1982).Google Scholar
8. Huppert, B., Endliche Gruppen I (Springer, Berlin/Heidelberg, 1967).CrossRefGoogle Scholar
9. Humphreys, J., Conjugacy classes in semisimple algebraic groups (Amer. Math. Soc, Providence, RI, 1995).Google Scholar
10. Jones, G., ‘Ree groups and Riemann surfaces’, J. Algebra 165 (1994), 41–62.CrossRefGoogle Scholar
11. Kleidman, P. and Liebeck, M. W., Subgroup structure of classical groups (Cambridge Univ. Press, Cambridge 1990).10.1017/CBO9780511629235CrossRefGoogle Scholar
12. Lucchini, A., Tamburini, M. C. and Wilson, J., Hurwitz groups of large rank, J. London Math. Soc. (2) 61 (2000) 81–92.10.1112/S0024610799008467CrossRefGoogle Scholar
13. Lucchini, A. and Tamburini, M. C., ‘Classical groups of large rank as Hurwitz groups’, J. Algebra 219 (1999) 531–546.10.1006/jabr.1999.7911CrossRefGoogle Scholar
14. Macbeath, A. M., Generators of the linear fractional groups, Proc. Symp. Pure Math. 12 (1969) 14–32.10.1090/pspum/012/0262379CrossRefGoogle Scholar
15. Malle, G., ‘Hurwitz groups and G2(q)’, Canad. Math. Bull. 33 (1990) 349–356.CrossRefGoogle Scholar
16. Malle, G., ‘Small rank exceptional Hurwitz groups’, Groups of Lie type and their geometries, London Math. Soc. Lecture Notes 207 (Cambridge Univ. Press, Cambridge, 1995) 173–183.CrossRefGoogle Scholar
17. Simpson, C. T., ‘Products of matrices’, Differential geometry, global analysis and topology, Canad. Math. Soc. Conf. Proc. 12 (Amer. Math. Soc, Providence, RI, 1991) 157–185.Google Scholar
18. Steinberg, R., ‘Lectures on Chevalley groups’, Yale Univ., 1967.Google Scholar
19. Scott, L., Matrices and cohomology, Ann. of Math. 105 (1987) 473–492.10.2307/1970920CrossRefGoogle Scholar
20. Springer, T. and Steinberg, R., ‘Conjugacy classes’, ‘Seminar on algebraic groups and related finite groups, Lect. Notes in Math. 131 (Springer, Berlin, 1970).10.1007/BFb0081546CrossRefGoogle Scholar
21. Strambach, K. and Völklein, H., ‘On linearly rigid tuples’, J. Reine Angew. Math. 510 (1999) 57–62.CrossRefGoogle Scholar
22. Tamburini, Ch. and Vassallo, S., ‘(2,3)-generazione di SL(4,q) in caracteristica dispari e problemi collegati, Boll Un. Mat. Ital. (7) (1994) 121–134.Google Scholar
23. Tamdurini, Ch. and Vsemirnov, M., ‘Hurwitz groups and Hurwitz generation’, Handbook of algebra, vol.4 (Elsevier, 2006) 385–426.10.1016/S1570-7954(06)80010-0CrossRefGoogle Scholar
24. Tamburini, Ch. and Vsemirnov, M., ‘Irreducible (2,3,7)-subgroups of PGLn(f), n ≤ 7’, J. Algebra 300 (2006) 339–362.10.1016/j.jalgebra.2006.02.030CrossRefGoogle Scholar
25. Chiara Tamburini, M. and Zalesski, A. E., ‘Classical groups in dimension 5 which are Hurwitz’, Finite groups 2003, Proc. Gainesville Conf. on Finite Groups, March 6–12, 2003 (ed. Ho, C. Y., Sin, P., Tiep, P. H. and Turull, A.; De Gruyter, Berlin, 2004) 363–372.Google Scholar
26. Vsemirnov, M., ‘Hurwitz groups of intermediate rank’, LMS J. Comput. Math. 7 (2004) 300–336.CrossRefGoogle Scholar
27. Wilson, R., ‘The Monster is a Hurwitz group’, J. Group Theory 4 (2001) 367–374.10.1515/jgth.2001.027CrossRefGoogle Scholar
28. Sun, Yongzhong, ‘On the (2,3,7)-Generation of some special linear groups’, Comm. Algebra 34 (2006) 51–74.10.1080/00927870500345935CrossRefGoogle Scholar
29. Zalesski, A., ‘Minimal polynomials and eigenvalues of p-elements in representations of groups with a cyclic Sylow p-subgroup’, J. London Math. Soc. 59 (1999) 845–866.CrossRefGoogle Scholar