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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Regular semisimple elements and involutions in finite general linear groups of odd characteristic


Authors: Cheryl E. Praeger and Ákos Seress
Journal: Proc. Amer. Math. Soc. 140 (2012), 3003-3015
MSC (2010): Primary 20D06, 20F69
Published electronically: January 25, 2012
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Abstract: Chris Parker and Rob Wilson showed that involution-centraliser methods could be used for solving several problems that appeared to be computationally hard and gave complexity analyses for methods to construct involutions and their centralisers in quasisimple Lie type groups in odd characteristic. Crucial to their analyses are conjugate involution pairs whose products are regular semisimple, possibly in an induced action on a subspace. We consider the fundamental case of conjugate involution pairs, in finite general linear groups $ \mathrm {GL}(n,q)$ with $ q$ odd, for which the product is regular semisimple on the underlying vector space. Such involutions form essentially a single conjugacy class $ \mathcal {C}$. We prove that a constant proportion of pairs from $ \mathcal {C}$ have regular semisimple product. Moreover we show that for a fixed parity of $ n$, this proportion converges exponentially quickly to a limit, as $ n$ approaches $ \infty $, the limit being $ (1-q^{-1})^2\Phi (q)^3$ for even $ n$ and $ (1-q^{-1})\Phi (q)^3$ for odd $ n$, where $ \Phi (q)=\prod _{i=1}^\infty (1-q^{-i})$.


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Additional Information

Cheryl E. Praeger
Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
Email: cheryl.praeger@uwa.edu.au

Ákos Seress
Affiliation: School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia – and – Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: akos@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11491-1
PII: S 0002-9939(2012)11491-1
Received by editor(s): March 28, 2011
Published electronically: January 25, 2012
Additional Notes: This work was partially supported by ARC Grants FF0776186 and DP1096525 and by the NSF
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.