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Geometric characterizations of finite Chevalley groups of type B$_{2}$
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by Koen Thas and Hendrik Van Maldeghem PDF
Trans. Amer. Math. Soc. 360 (2008), 2327-2357 Request permission

Abstract:

Finite Moufang generalized quadrangles were classified in 1974 as a corollary to the classification of finite groups with a split BN-pair of rank $2$, by P. Fong and G. M. Seitz (1973), (1974). Later on, work of S. E. Payne and J. A. Thas culminated in an almost complete, elementary proof of that classification; see Finite Generalized Quadrangles, 1984. Using slightly more group theory, first W. M. Kantor (1991), then the first author (2001), and finally, essentially without group theory, J. A. Thas (preprint), completed this geometric approach. Recently, J. Tits and R. Weiss classified all (finite and infinite) Moufang polygons (2002), and this provides a third independent proof for the classification of finite Moufang quadrangles. In the present paper, we start with a much weaker condition on a BN-pair of Type $\mathbf {B}_2$ and show that it must correspond to a Moufang quadrangle, proving that the BN-pair arises from a finite Chevalley group of (relative) Type $\mathbf {B}_2$. Our methods consist of a mixture of combinatorial, geometric and group theoretic arguments, but we do not use the classification of finite simple groups. The condition on the BN-pair translates to the generalized quadrangle as follows: for each point $x$, the stabilizer of all lines through that point acts transitively on the points opposite $x$.
References
  • I. Bloemen, J. A. Thas, and H. Van Maldeghem, Elation generalized quadrangles of order $(p,t)$, $p$ prime, are classical, J. Statist. Plann. Inference 56 (1996), no. 1, 49–55. Special issue on orthogonal arrays and affine designs, Part I. MR 1435520, DOI 10.1016/S0378-3758(97)83059-5
  • R. C. Bose and S. S. Shrikhande, Geometric and pseudo-geometric graphs $(q^{2}+1,\,q+1,\,1)$, J. Geom. 2 (1972), 75–94. MR 302468, DOI 10.1007/BF02148139
  • A. E. Brouwer, The complement of a geometric hyperplane in a generalized polygon is usually connected, Finite geometry and combinatorics (Deinze, 1992) London Math. Soc. Lecture Note Ser., vol. 191, Cambridge Univ. Press, Cambridge, 1993, pp. 53–57. MR 1256263, DOI 10.1017/CBO9780511526336.007
  • F. Buekenhout and H. Van Maldeghem, Finite distance-transitive generalized polygons, Geom. Dedicata 52 (1994), no. 1, 41–51. MR 1296145, DOI 10.1007/BF01263523
  • 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
  • Paul Fong and Gary M. Seitz, Groups with a $(B,\,N)$-pair of rank $2$, Finite groups ’72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972) North-Holland Math. Studies, Vol. 7, North-Holland, Amsterdam, 1973, pp. 36–40. MR 0360796, DOI 10.1007/BF01390051
  • 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
  • Christoph Hering, William M. Kantor, and Gary M. Seitz, Finite groups with a split $BN$-pair of rank $1$. I, J. Algebra 20 (1972), 435–475. MR 301085, DOI 10.1016/0021-8693(72)90068-3
  • D. G. Higman, Partial geometries, generalized quadrangles and strongly regular graphs, Atti del Convegno di Geometria Combinatoria e sue Applicazioni (Univ. Perugia, Perugia, 1970) Ist. Mat., Univ. Perugia, Perugia, 1971, pp. 263–293. MR 0366698
  • D. G. Higman, Invariant relations, coherent configurations and generalized polygons, Combinatorics (Proc. NATO Advanced Study Inst., Breukelen, 1974) Math. Centre Tracts, No. 57, Math. Centrum, Amsterdam, 1974, pp. 27–43. MR 0379244
  • J. W. P. Hirschfeld, Projective geometries over finite fields, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 554919
  • B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3
  • 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
  • Donald Passman, Permutation groups, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0237627
  • Stanley E. Payne, Finite generalized quadrangles: a survey, Proceedings of the International Conference on Projective Planes (Washington State Univ., Pullman, Wash., 1973) Washington State Univ. Press, Pullman, Wash., 1973, pp. 219–261. MR 0363954
  • Stanley E. Payne, Collineations of finite generalized quadrangles, Finite geometries (Pullman, Wash., 1981) Lecture Notes in Pure and Appl. Math., vol. 82, Dekker, New York, 1983, pp. 361–390. MR 690820
  • 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
  • Gary M. Seitz, Flag-transitive subgroups of Chevalley groups, Ann. of Math. (2) 97 (1973), 27–56. MR 340446, DOI 10.2307/1970876
  • Ernest Shult, On a class of doubly transitive groups, Illinois J. Math. 16 (1972), 434–445. MR 296150
  • Katrin Tent and Hendrik Van Maldeghem, Moufang polygons and irreducible spherical BN-pairs of rank 2. I, Adv. Math. 174 (2003), no. 2, 254–265. MR 1963695, DOI 10.1016/S0001-8708(02)00039-7
  • J. A. Thas, Characterizations of generalized quadrangles by generalized homologies, J. Combin. Theory Ser. A 40 (1985), no. 2, 331–341. MR 814418, DOI 10.1016/0097-3165(85)90094-9
  • J. A. Thas, The classification of all $(x,y)$-transitive generalized quadrangles, J. Combin. Theory Ser. A 42 (1986), no. 1, 154–157. MR 843472, DOI 10.1016/0097-3165(86)90016-6
  • J. A. Thas. Generalized quadrangles and the theorem of Fong and Seitz on BN-pairs, Preprint.
  • J. A. Thas and H. Van Maldeghem, Lax embeddings of generalized quadrangles in finite projective spaces, Proc. London Math. Soc. (3) 82 (2001), no. 2, 402–440. MR 1806877, DOI 10.1112/S0024611501012680
  • J. A. Thas, S. E. Payne, and Van Maldeghem, Half Moufang implies Moufang for finite generalized quadrangles, Invent. Math. 105 (1991), no. 1, 153–156. MR 1109623, DOI 10.1007/BF01232260
  • K. Thas. Strong Elation Generalized Quadrangles, I and II, Lectures given at Ghent University Incidence Geometry Seminar, 1999.
  • Koen Thas, On symmetries and translation generalized quadrangles, Finite geometries, Dev. Math., vol. 3, Kluwer Acad. Publ., Dordrecht, 2001, pp. 333–345. MR 2061813, DOI 10.1007/978-1-4613-0283-4_{2}0
  • K. Thas. Automorphisms and Characterizations of Finite Generalized Quadrangles, in: Generalized Polygons, Proceedings of the Academy Contact Forum ‘Generalized Polygons’, 20 October 2000, Palace of the Academies, Brussels (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
  • Koen Thas, Classification of span-symmetric generalized quadrangles of order $s$, Adv. Geom. 2 (2002), no. 2, 189–196. MR 1895346, DOI 10.1515/advg.2002.005
  • Koen Thas, The classification of generalized quadrangles with two translation points, Beiträge Algebra Geom. 43 (2002), no. 2, 365–398. MR 1957745
  • K. Thas. Automorphisms and Combinatorics of Finite Generalized Quadrangles, Ph.D. Thesis, Ghent University, March 2002.
  • Koen Thas, Symmetry in finite generalized quadrangles, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2004. MR 2036432
  • J. Tits. Sur la trialité et certains groupes qui s’en déduisent, Inst. Hautes Etudes Sci. Publ. Math. 2 (1959), 13–60.
  • J. Tits. Les groupes simples de Suzuki et de Ree, Séminaire Bourbaki 13(210) (1960), 1 – 18.
  • Jacques Tits, Théorème de Bruhat et sous-groupes paraboliques, C. R. Acad. Sci. Paris 254 (1962), 2910–2912 (French). MR 138706
  • Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
  • J. Tits, Classification of buildings of spherical type and Moufang polygons: a survey, Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973) Atti dei Convegni Lincei, No. 17, Accad. Naz. Lincei, Rome, 1976, pp. 229–246 (English, with Italian summary). MR 0444793
  • Jacques Tits, Twin buildings and groups of Kac-Moody type, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 249–286. MR 1200265, DOI 10.1017/CBO9780511629259.023
  • 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, A geometric characterization of the perfect Suzuki-Tits ovoids, J. Geom. 58 (1997), no. 1-2, 192–202. MR 1434192, DOI 10.1007/BF01222940
  • Hendrik Van Maldeghem, Some consequences of a result of Brouwer, Ars Combin. 48 (1998), 185–190. MR 1623011
  • Hendrik van Maldeghem, Generalized polygons, Monographs in Mathematics, vol. 93, Birkhäuser 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, Galglaan 2, B-9000 Ghent, Belgium
  • Email: kthas@cage.UGent.be
  • Hendrik Van Maldeghem
  • Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Ghent, Belgium
  • Email: hvm@cage.UGent.be
  • Received by editor(s): January 18, 2005
  • Received by editor(s) in revised form: December 28, 2005
  • Published electronically: November 20, 2007
  • Additional Notes: The first author is a Postdoctoral Fellow of the Fund for Scientific Research — Flanders (Belgium).
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2327-2357
  • MSC (2000): Primary 05B25, 51E12, 20B10, 20B25, 20E42
  • DOI: https://doi.org/10.1090/S0002-9947-07-04257-2
  • MathSciNet review: 2373316