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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Solution of a question of Knarr
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by Koen Thas PDF
Proc. Amer. Math. Soc. 136 (2008), 1409-1418 Request permission

Abstract:

The Moufang condition is one of the central group theoretical conditions in Incidence Geometry, and was introduced by Jacques Tits in his famous lecture notes (1974). About ten years ago, Norbert Knarr studied generalized quadrangles (buildings of Type $\mathbf {B}_2$) which satisfy one of the Moufang conditions locally at one point. He then posed the fundamental question whether the group generated by the root-elations with its root containing that point is always a sharply transitive group on the points opposite this point, that is, whether this group is an elation group. In this paper, we solve the question and a more general version affirmatively for finite generalized quadrangles. Moreover, we show that this group is necessarily nilpotent (which was only known up till now when both Moufang conditions are satisfied for all points and lines). In fact, as a corollary, we will prove that these groups always have to be $p$-groups for some prime $p$.
References
  • Paul Fong and Gary M. Seitz, Groups with a $(B,\,N)$-pair of rank $2$. I, II, Invent. Math. 21 (1973), 1–57; ibid. 24 (1974), 191–239. MR 354858, DOI 10.1007/BF01389689
  • Daniel Frohardt, Groups which produce generalized quadrangles, J. Combin. Theory Ser. A 48 (1988), no. 1, 139–145. MR 938864, DOI 10.1016/0097-3165(88)90081-7
  • Daniel Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co., New York, 1980. MR 569209
  • Dirk Hachenberger, Groups admitting a Kantor family and a factorized normal subgroup, Des. Codes Cryptogr. 8 (1996), no. 1-2, 135–143. Special issue dedicated to Hanfried Lenz. MR 1393979, DOI 10.1007/BF00130573
  • Daniel R. Hughes and Fred C. Piper, Projective planes, Graduate Texts in Mathematics, Vol. 6, Springer-Verlag, New York-Berlin, 1973. MR 0333959
  • W. M. Kantor, Automorphism groups of some generalized quadrangles, Advances in finite geometries and designs (Chelwood Gate, 1990) Oxford Sci. Publ., Oxford Univ. Press, New York, 1991, pp. 251–256. MR 1138747
  • N. Knarr. Private Communication, October 2004 and June 2005.
  • S. E. Payne and J. A. Thas, Finite generalized quadrangles, Research Notes in Mathematics, vol. 110, Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 767454
  • Stanley E. Payne and Koen Thas, Notes on elation generalized quadrangles, European J. Combin. 24 (2003), no. 8, 969–981. MR 2024555, DOI 10.1016/j.ejc.2003.07.002
  • K. Thas. Automorphisms and Characterizations of Finite Generalized Quadrangles, in: Generalized Polygons, Proceedings of the Academy Contact Forum “Generalized Polygons” 20 October, Palace of the Academies, Brussels, Belgium (2001), 111–172.
  • Koen Thas, A theorem concerning nets arising from generalized quadrangles with a regular point, Des. Codes Cryptogr. 25 (2002), no. 3, 247–253. MR 1900771, DOI 10.1023/A:1014931328755
  • K. Thas. Automorphisms and Combinatorics of Finite Generalized Quadrangles, Ph.D. Thesis, Ghent University, Ghent (2002), xxviii+412 pp.
  • Koen Thas, Symmetry in generalized quadrangles, Proceedings of the Conference on Finite Geometries (Oberwolfach, 2001), 2003, pp. 227–245. MR 1993170, DOI 10.1023/A:1024168928527
  • Koen Thas, Symmetry in finite generalized quadrangles, Frontiers in Mathematics, BirkhĂ€user Verlag, Basel, 2004. MR 2036432
  • Koen Thas, Some basic questions and conjectures on elation generalized quadrangles, and their solutions, Bull. Belg. Math. Soc. Simon Stevin 12 (2005), no. 5, 909–918. MR 2241353
  • Koen Thas and Stanley E. Payne, Foundations of elation generalized quadrangles, European J. Combin. 27 (2006), no. 1, 51–62. MR 2186415, DOI 10.1016/j.ejc.2004.08.001
  • Koen Thas and Hendrik Van Maldeghem, Moufang-like conditions for generalized quadrangles and classification of all finite quasi-transitive generalized quadrangles, Discrete Math. 294 (2005), no. 1-2, 203–217. MR 2139798, DOI 10.1016/j.disc.2004.04.047
  • K. Thas and H. Van Maldeghem. Geometric characterizations of finite Chevalley groups of Type $\mathbf {B}_2$, Trans. Amer. Math. Soc., To appear.
  • Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
  • Jacques Tits and Richard M. Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1938841, DOI 10.1007/978-3-662-04689-0
  • Hendrik van Maldeghem, Generalized polygons, Monographs in Mathematics, vol. 93, BirkhĂ€user Verlag, Basel, 1998. MR 1725957, DOI 10.1007/978-3-0348-0271-0
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Additional Information
  • Koen Thas
  • Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium
  • Email: kthas@cage.UGent.be
  • Received by editor(s): June 10, 2005
  • Received by editor(s) in revised form: October 31, 2006, and November 23, 2006
  • Published electronically: December 4, 2007
  • Additional Notes: The author is a Postdoctoral Fellow of the Fund for Scientific Research — Flanders (Belgium).
  • Communicated by: Jonathan I. Hall
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 136 (2008), 1409-1418
  • MSC (2000): Primary 51E12, 20B25, 20E42
  • DOI: https://doi.org/10.1090/S0002-9939-07-09136-8
  • MathSciNet review: 2367114