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

DOI:
https://doi.org/10.1090/S0002-9939-2012-11491-1

Published electronically:
January 25, 2012

MathSciNet review:
2917073

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Abstract | References | Similar Articles | Additional Information

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 with odd, for which the product is regular semisimple on the underlying vector space. Such involutions form essentially a single conjugacy class . We prove that a constant proportion of pairs from have regular semisimple product. Moreover we show that for a fixed parity of , this proportion converges exponentially quickly to a limit, as approaches , the limit being for even and for odd , where .

**1.**C. Altseimer and A. Borovik, Probabilistic recognition of orthogonal and symplectic groups. In:*Groups and Computation III*. Editors: W.M. Kantor and Á. Seress, de Gruyter, Berlin, New York (2001), pp. 1-20. With corrections in http://www.ma.umist.ac.uk/avb/pdf/alt-avb4.pdf. MR**1829468 (2002e:20093)****2.**J. N. Bray, An improved method for generating the centralizer of an involution,*Arch. Math. (Basel)***74**(2000), 241-245. MR**1742633 (2001c:20063)****3.**R. W. Carter,*Finite groups of Lie type: Conjugacy classes and complex characters*, Wiley, Chichester, 1993. MR**1266626 (94k:20020)****4.**J. E. Fulman, Peter M. Neumann and Cheryl E. Praeger,*A Generating Function Approach to the Enumeration of Matrices in Classical Groups over Finite Fields*, Memoirs of the American Mathematical Society**176**(2005), no. 830, vi+90 pp. MR**2145026 (2006b:05125)****5.**Simon Guest and Cheryl E. Praeger, Proportions of -part orders of elements in finite classical groups, submitted. Available at arxiv.org/abs/1007.2983**6.**B. Hartley and T. O. Hawkes,*Rings, modules and linear algebra. A further course in algebra describing the structure of abelian groups and canonical forms of matrices through the study of rings and modules*. A reprinting. Chapman & Hall, London-New York, 1980. MR**619212 (82e:00001)****7.**P. E. Holmes, S. A. Linton, E. A. O'Brien, A. J. E. Ryba, and R. A. Wilson, Constructive membership in black-box groups.*J. Group Theory***11**(2008), 747-763. MR**2466905 (2009i:20001)****8.**B. Huppert,*Endliche Gruppen I*, Springer, Berlin, 1967. MR**0224703 (37:302)****9.**C.R. Leedham-Green and E.A. O'Brien, Constructive recognition of classical groups in odd characteristic,*J. Algebra***322**(2009), 833-881. MR**2531225 (2010e:20075)****10.**M. W. Liebeck and E. A. O'Brien, Finding the characteristic of a group of Lie type,*J. London Math. Soc.***75**(2007), 741-754. MR**2352733 (2008i:20058)****11.**Frank Lübeck, Alice C. Niemeyer, and Cheryl E. Praeger, Finding involutions in finite Lie type groups of odd characteristic,*J. Algebra***321**(2009), 3397-3417. MR**2510054 (2010e:20026)****12.**Peter M. Neumann and Cheryl E. Praeger, Cyclic matrices over finite fields,*J. London Math. Soc. (2)***52**(1995), 263-284. MR**1356142 (96j:15017)****13.**Christopher W. Parker and Robert A. Wilson, Recognising simplicity of black-box groups by constructing involutions and their centralisers,*J. Algebra***324**(2010), 886-915. MR**2659204 (2011j:20032)****14.**Cheryl E. Praeger and Ákos Seress, Probabilistic generation of finite classical groups in odd characteristic by involutions,*J. Group Theory***14**(2011), 521-545. MR**2818948**

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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:
https://doi.org/10.1090/S0002-9939-2012-11491-1

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.