Hostname: page-component-6766d58669-l4t7p Total loading time: 0 Render date: 2026-05-16T12:58:07.137Z Has data issue: false hasContentIssue false
  • English
  • Français

ALGORITHMS TO IDENTIFY ABUNDANT p-SINGULAR ELEMENTS IN FINITE CLASSICAL GROUPS

Part of: Linear algebraic groups and related topics

Published online by Cambridge University Press:  06 August 2012

ALICE C. NIEMEYER ,
TOMASZ POPIEL  and
CHERYL E. PRAEGER
ALICE C. NIEMEYER
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (email: alice.niemeyer@uwa.edu.au)
TOMASZ POPIEL
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (email: tomasz.popiel@uwa.edu.au)
CHERYL E. PRAEGER*
Affiliation:
The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia King Abdulaziz University, Jeddah, Saudi Arabia (email: cheryl.praeger@uwa.edu.au)
*
✉For correspondence; e-mail: cheryl.praeger@uwa.edu.au
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.

Let G be a finite d-dimensional classical group and p a prime divisor of ∣G∣ distinct from the characteristic of the natural representation. We consider a subfamily of p-singular elements in G (elements with order divisible by p) that leave invariant a subspace X of the natural G-module of dimension greater than d/2and either act irreducibly on X or preserve a particular decomposition of X into two equal-dimensional irreducible subspaces. We proved in a recent paper that the proportion in G of these so-called p-abundant elements is at least an absolute constant multiple of the best currently known lower bound for the proportion of all p-singular elements. From a computational point of view, the p-abundant elements generalise another class of p-singular elements which underpin recognition algorithms for finite classical groups, and it is our hope that p-abundant elements might lead to improved versions of these algorithms. As a step towards this, here we present efficient algorithms to test whether a given element is p-abundant, both for a known prime p and for the case where p is not known a priori.

Keywords

finite groups of Lie typeproportions of elementslarge invariant subspaces

MSC classification

Secondary: 20G40: Linear algebraic groups over finite fields

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 1 , August 2012 , pp. 50 - 63
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

The first and third authors acknowledge the support of the Australian Research Council Discovery Project DP110101153. The second author acknowledges support within the Australian Research Council Federation Fellowship FF0776186 of the third author.

References

[1]Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: the user language’, J. Symbolic Comput. 24 (1997), 235–265.Google Scholar
[2]Coppersmith, D. and Winograd, S., ‘Matrix multiplication via arithmetic progressions’, J. Symbolic Comput. 9 (1990), 251–280.CrossRefGoogle Scholar
[3]Gambini Weigel, A. and Weigel, T. S., ‘On the orders of primitive linear p′-groups’, Bull. Aust. Math. Soc. 48 (1993), 495–521.CrossRefGoogle Scholar
[4] ‘GAP—groups, algorithms, programming—a system for computational discrete algebra’,http://www.gap-system.org/.Google Scholar
[5]Guest, S. and Praeger, C. E., ‘Proportions of elements with given 2-part order in finite classical groups of odd characteristic’, Preprint, 2011, http://arxiv.org/abs/1007.2983v4.CrossRefGoogle Scholar
[6]Guralnick, R. M. and Lübeck, F., ‘On p-singular elements in Chevalley groups in characteristic p’, in: Groups and Computation III, Ohio State University Mathematical Research Institute Publications, 8 (de Gruyter, Berlin, 2001), pp. 169–182.Google Scholar
[7]Hartley, B. and Hawkes, T. O., Rings, Modules and Linear Algebra (Chapman and Hall, London, 1970).Google Scholar
[8]Huppert, B., Endliche Gruppen I, Die Grundlehren der Mathematischen Wissenschaften, 134 (Springer, Berlin, 1967).Google Scholar
[9]Huppert, B., ‘Singer-Zyklen in klassischen Gruppen’, Math. Z. 117 (1970), 141–150.CrossRefGoogle Scholar
[10]Huppert, B., ‘Isometrien von Vektorräumen 1’, Arch. Math. 35 (1980), 164–176.CrossRefGoogle Scholar
[11]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.Google Scholar
[12]Keller-Gehrig, W., ‘Fast algorithms for the characteristics polynomial’, Theoret. Comput. Sci. 36 (1985), 309–317.CrossRefGoogle Scholar
[13]Lübeck, F., ‘Finding p′-elements in finite groups of Lie type’, in: Groups and Computation III, Ohio State University Mathematical Research Institute Publications, 8 (de Gruyter, Berlin, 2001), pp. 249–256.CrossRefGoogle Scholar
[14]Möller, N., ‘On Schönhage’s algorithm and subquadratic integer gcd computation’, Math. Comp. 77 (2008), 589–607.CrossRefGoogle Scholar
[15]Neumann, P. M. and Praeger, C. E., ‘A recognition algorithm for special linear groups’, Proc. Lond. Math. Soc. (3) 65 (1992), 555–603.CrossRefGoogle Scholar
[16]Niemeyer, A. C., Popiel, T. and Praeger, C. E., ‘Abundant p-singular elements in finite classical groups’, Preprint, 2012, http://arxiv.org/abs/1205.1454.Google Scholar
[17]Niemeyer, A. C. and Praeger, C. E., ‘Implementing a recognition algorithm for classical groups’, in: Groups and Computation, II, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 28 (American Mathematical Society, Providence, RI, 1997), pp. 273–296.Google Scholar
[18]Niemeyer, A. C. and Praeger, C. E., ‘A recognition algorithm for classical groups over finite fields’, Proc. Lond. Math. Soc. (3) 77 (1998), 117–169.CrossRefGoogle Scholar
[19]Niemeyer, A. C. and Praeger, C. E., ‘Estimating proportions of elements in finite groups of Lie type’, J. Algebra 324 (2010), 122–145.CrossRefGoogle Scholar
[20]Niven, I., Zuckerman, H. S. and Montgomery, H. L., An Introduction to the Theory of Numbers, 5th edn (John Wiley & Sons, New York, NY, 1991).Google Scholar
[21]Praeger, C. E., ‘Primitive prime divisor elements in finite classical groups’, in: Groups St. Andrews 1997 in Bath, II, London Mathematical Society Lecture Note Series, 261 (Cambridge University Press, Cambridge, 1999), pp. 605–623.CrossRefGoogle Scholar
[22]Schönhage, A. and Strassen, V., ‘Schnelle Multiplikation großer Zahlen’, Computing 7 (1971), 281–292.CrossRefGoogle Scholar