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), 30033015
MSC (2010):
Primary 20D06, 20F69
Published electronically:
January 25, 2012
MathSciNet review:
2917073
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Abstract 
References 
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Additional Information
Abstract: Chris Parker and Rob Wilson showed that involutioncentraliser 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.
Christine
Altseimer and Alexandre
V. Borovik, Probabilistic recognition of orthogonal and symplectic
groups, Groups and computation, III (Columbus, OH, 1999) Ohio State
Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001,
pp. 1–20. MR 1829468
(2002e:20093)
 2.
John
N. Bray, An improved method for generating the centralizer of an
involution, Arch. Math. (Basel) 74 (2000),
no. 4, 241–245. MR 1742633
(2001c:20063), 10.1007/s000130050437
 3.
Roger
W. Carter, Finite groups of Lie type, Wiley Classics Library,
John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and
complex characters; Reprint of the 1985 original; A WileyInterscience
Publication. MR
1266626 (94k:20020)
 4.
Jason
Fulman, Peter
M. Neumann, and Cheryl
E. Praeger, A generating function approach to the enumeration of
matrices in classical groups over finite fields, Mem. Amer. Math. Soc.
176 (2005), no. 830, vi+90. MR 2145026
(2006b:05125), 10.1090/memo/0830
 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, Chapman &
Hall, LondonNew York, 1980. 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. 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 blackbox groups, J.
Group Theory 11 (2008), no. 6, 747–763. MR 2466905
(2009i:20001), 10.1515/JGT.2008.047
 8.
B.
Huppert, Endliche Gruppen. I, Die Grundlehren der
Mathematischen Wissenschaften, Band 134, SpringerVerlag, BerlinNew York,
1967 (German). MR 0224703
(37 #302)
 9.
C.
R. LeedhamGreen and E.
A. O’Brien, Constructive recognition of classical groups in
odd characteristic, J. Algebra 322 (2009),
no. 3, 833–881. MR 2531225
(2010e:20075), 10.1016/j.jalgebra.2009.04.028
 10.
Martin
W. Liebeck and E.
A. O’Brien, Finding the characteristic of a group of Lie
type, J. Lond. Math. Soc. (2) 75 (2007), no. 3,
741–754. MR 2352733
(2008i:20058), 10.1112/jlms/jdm028
 11.
Frank
Lübeck, Alice
C. Niemeyer, and Cheryl
E. Praeger, Finding involutions in finite Lie type groups of odd
characteristic, J. Algebra 321 (2009), no. 11,
3397–3417. MR 2510054
(2010e:20026), 10.1016/j.jalgebra.2008.05.009
 12.
Peter
M. Neumann and Cheryl
E. Praeger, Cyclic matrices over finite fields, J. London
Math. Soc. (2) 52 (1995), no. 2, 263–284. MR 1356142
(96j:15017), 10.1112/jlms/52.2.263
 13.
Christopher
W. Parker and Robert
A. Wilson, Recognising simplicity of blackbox groups by
constructing involutions and their centralisers, J. Algebra
324 (2010), no. 5, 885–915. MR 2659204
(2011j:20032), 10.1016/j.jalgebra.2010.05.013
 14.
Cheryl
E. Praeger and Ákos
Seress, Probabilistic generation of finite classical groups in odd
characteristic by involutions, J. Group Theory 14
(2011), no. 4, 521–545. MR 2818948
(2012g:20131), 10.1515/JGT.2010.061
 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. 120. With corrections in http://www.ma.umist.ac.uk/avb/pdf/altavb4.pdf. MR 1829468 (2002e:20093)
 2.
 J. N. Bray, An improved method for generating the centralizer of an involution, Arch. Math. (Basel) 74 (2000), 241245. 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, LondonNew 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 blackbox groups. J. Group Theory 11 (2008), 747763. MR 2466905 (2009i:20001)
 8.
 B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967. MR 0224703 (37:302)
 9.
 C.R. LeedhamGreen and E.A. O'Brien, Constructive recognition of classical groups in odd characteristic, J. Algebra 322 (2009), 833881. 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), 741754. 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), 33973417. MR 2510054 (2010e:20026)
 12.
 Peter M. Neumann and Cheryl E. Praeger, Cyclic matrices over finite fields, J. London Math. Soc. (2) 52 (1995), 263284. MR 1356142 (96j:15017)
 13.
 Christopher W. Parker and Robert A. Wilson, Recognising simplicity of blackbox groups by constructing involutions and their centralisers, J. Algebra 324 (2010), 886915. 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), 521545. 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.ohiostate.edu
DOI:
http://dx.doi.org/10.1090/S000299392012114911
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.
