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On the Number of p-Regular Elements in Finite Simple Groups

Published online by Cambridge University Press:  01 February 2010

László Babai ,
Péter P. Pálfy  and
Jan Saxl
László Babai
Affiliation:
Department of Computer Science, University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA, [email protected]
Péter P. Pálfy
Affiliation:
Rényi Institute and Eötvös University, 13–15 Reáltanoda u., 1053 Budapest, Hungary, [email protected]
Jan Saxl
Affiliation:
Gonville and Caius College, Trinity Street, Cambridge CB2 1TA, United Kingdom, [email protected]

Abstract

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A p-regular element in a finite group is an element of order not divisible by the prime number p. We show that for every prime p and every finite simple group S, a fair proportion of elements of S is p-regular. In particular, we show that the proportion of p-regular elements in a finite classical simple group (not necessarily of characteristic p) is greater than 1/(2n), where n – 1 is the dimension of the projective space on which S acts naturally. Furthermore, in an exceptional group of Lie type this proportion is greater than 1/15. For the alternating group An, this proportion is at least 26/(27√n), and for sporadic simple groups, at least 2/29.

We also show that for an arbitrary field F, if the simple group S is a quotient of a finite subgroup of GLn(F) then for any prime p, the proportion of p-regular elements in S is at least min{1/31, 1/(2n)}.

Along the way we obtain estimates for the proportion of elements of certain primitive prime divisor orders in exceptional groups, complementing work by Niemeyer and Praeger (1998).Our result shows that in finite simple groups, p-regular elements can be found efficiently by random sampling. This is a key ingredient to recent polynomial-time Monte Carlo algorithms for matrix groups.

Finally we complement our lower bound results with the following upper bound: for all n ≥ 2 there exist infinitely many prime powers q such that the proportion of elements of odd order in PSL(n,q) is less than 3/√n.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 12 , 2009 , pp. 82 - 119
Copyright
Copyright © London Mathematical Society 2009

References

1.Babai, L. and Szemerédi, E., ‘On the complexity of matrix group problems I’, Proc. 25th IEEE Symp. on Theory of Computing (FOCS'84) (IEEE Comp. Soc. Press, 1984) 229–240.CrossRefGoogle Scholar
2.Babai, L. and Beals, R., ‘A polynomial-time theory of black box groups I’, Groups St Andrews 1997 in Bath, I (ed. Campbell, C. M. et al. ), London Math. Soc. Lecture Note Ser. 260 (Cambridge University Press, Cambridge, 1999) 30–64.CrossRefGoogle Scholar
3.Babai, L., Beals, R. and Seress, Á., ‘Polynomial-time theory of matrix groups’, Proc. 41st ACM Symp. Theory of Computing (STOC'09) (ACM Press), to appear.Google Scholar
4.Babai, L. and Shalev, A., ‘Recognizing simplicity of black-box groups and the frequency of p-sigular elements in affine groups’, Groups and Computation, (ed. Kantor, W. M. and Seress, Á.), Ohio State Univ. Math. Res. Inst. Publ. 8 (de Gruyter, Berlin, 2001) 39–62.CrossRefGoogle Scholar
5.Beals, R., Leedham-Green, C., Niemeyer, A. C., Praeger, C. E. and Seress, Á., ‘Permutations with restricted cycle structure and an algorithmic application’, Combinatorics, Probability and Computing 11 (2002) 446–464.CrossRefGoogle Scholar
6.Carter, R., ‘Conjugacy classes in the Weyl group’, in Seminar on algebraic groups and related finite groups (ed. Borel, A. et al. ), Lecture Notes in Mathematics 131 (Springer, Berlin, 1970) 297–318.CrossRefGoogle Scholar
7.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
8.Erdős, P. and Turán, P., ‘On some problems of a statistical group theory II,’ Acta Math. Acad. Sci. Hungar. 18 (1967) 151–163.CrossRefGoogle Scholar
9.Feit, W. and Tits, J., ‘Projective representations of minimum degree of group extensions’, Canad. J. Math. 30 (1978) 1092–1102.CrossRefGoogle Scholar
10.Fulman, J., Neumann, P. M. and Praeger, C. E., ‘A Generating Function Approach to the Enumeration of Matrices in Groups over Finite Fields’, Mem. Amer. Math. Soc. 176, no. 830 (2005).Google Scholar
11.Gager, P. C., ‘Maximal tori in finite groups of Lie type’, PhD Thesis, University of Warwick, 1973.Google Scholar
12.Guralnick, R. M. and Lübeck, F., ‘On p-singular elements in Chevalley groups in characteristic p’, Groups and Computation III, (ed. Kantor, W. M. and Seress, Á.), Ohio State Univ. Math. Res. Inst. Publ. 8 (de Gruyter, Berlin, 2001) 169–182.CrossRefGoogle Scholar
13.Huppert, B., ‘Singer-Zyklen in klassischen Gruppen’, Math. Z. 117 (1970) 141–150.CrossRefGoogle Scholar
14.Isaacs, I. M., Kantor, W. M. and Spaltenstein, N., ‘On the probability that a group element is p-singular’, J. Algebra 176 (1995) 139–181.CrossRefGoogle Scholar
15.Kantor, W. M. and Seress, Á., ‘Prime power graphs for groups of Lie type’, J. Algebra 247 (2002) 370–434.CrossRefGoogle Scholar
16.Kleibman, P. and Liebeck, M., The subgroup structure of the finite classi cal groups, London Math. Soc. Lecture Note Ser. 129 (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
17.Lawther, R., personal communication, April 2008.Google Scholar
18.Landazuri, V. and Seitz, G. M., ‘On the minimal degrees of projective representations of the finite Chevalley groups’, J. Algebra 32 (1974) 418–443.CrossRefGoogle Scholar
19.Liebeck, M. W., Saxl, J. and Seitz, G. M., ‘Subgroups of maximal rank in finite exceptional groups of Lie type’, Proc. London Math. Soc. 65 (1992) 297–325.CrossRefGoogle Scholar
20.Malle, G., Saxl, J. and Weigel, T., ‘Generation of classical groups’, Geom. Dedicata 49 (1994) 85–116.CrossRefGoogle Scholar
21.Maróti, A., ‘Symmetric functions, generalized blocks and permutations with restricted cycle structure’, European J. Combin. 28 (2007) 942–963.CrossRefGoogle Scholar
22.Neumann, P. M. and Praeger, C. E., ‘Cyclic matrices in classical groups over finite fields’, J. Algebra 234 (2000) 367–418.CrossRefGoogle Scholar
23.Niemeyer, A. C. and Praeger, C. E., ‘A recognition algorithm for classical groups over finite fields’, Proc. London Math. Soc. 77 (1998) 117–169.CrossRefGoogle Scholar
24.Niemeyer, A. C. and Praeger, C. E., ‘A recognition algorithm for non-generic classical groups over finite fields’, J. Austral. Math. Soc. Ser. A 67 (1999) 223–253.CrossRefGoogle Scholar
25.Springer, T. A. and Steinberg, R., ‘Conjugacy classes’, in Seminar on algebraic groups and related finite groups (ed. Borel, A. et al. ), Lecture Notes in Mathematics 131 (Springer, Berlin, 1970) 167–266.CrossRefGoogle Scholar
26.Taylor, D. E., The geometry of the classical groups, Sigma Series in Pure Math. 9 (Heldermann, Berlin, 1992).Google Scholar
27.Zsigmondy, K., ‘Zur Theorie der Potenzreste’, Monatsh. Math. Phys. 3 (1892) 265–284.CrossRefGoogle Scholar