Generalized polygons, SCABs and GABs

  • William M. Kantor
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 1181)


Automorphism Group Chevalley Group Frobenius Group Generalize Quadrangle Chamber System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A 1]
    M. Aschbacher, Flag structures on Tits geometries. Geom. Ded. 14 (1983), 21–32.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [A 2]
    ___, Finite geometries of type C3 with flag transitive groups. Geom. Ded. 16 (1984), 195–200.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [AS]
    ___ and S.D. Smith, Tits geometries over GF(2) defined by groups over GF(3). Comm. in Alg. 11 (1983), 1675–1684.CrossRefMathSciNetGoogle Scholar
  4. [ASz]
    R.W. Ahrens and G. Szekeres, On a combinatorial generalization of 27 lines associated with a cubic surface. J. Austral. Math. Soc. 10 (1969), 485–492.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [BCT]
    A.E. Brouwver, A.M. Cohen and J. Tits, Some remarks on Tits geometries. Indag. Math. 45 (1983), 393–400.Google Scholar
  6. [Bi]
    N.L. Biggs, Algebraic graph theory. Cambridge University Press, Cambridge 1974.zbMATHCrossRefGoogle Scholar
  7. [BT]
    F. Bruhat and J. Tits, Groupes réductifs sur un corps local. I. Données radicielles valuées. Publ. Math. I.H.E.S. 41 (1972), 5–251.zbMATHMathSciNetGoogle Scholar
  8. [Bu]
    F. Buekenhout, Diagrams for geometries and groups. JCT(A) 27 (1979), 121–151.zbMATHMathSciNetGoogle Scholar
  9. [Ca]
    P.J. Cameron, Partial quadrangles. Quart. J. Math. 26 (1974), 61–73.CrossRefGoogle Scholar
  10. [CaK]
    ___ and W.M. Kantor, 2-Transitive and antiflag transitive collineation groups of finite projective and polar spaces. J. Alg. 60 (1979), 384–422.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [Car]
    R.W. Carter, Simple groups of Lie type. Wiley, New York 1972.zbMATHGoogle Scholar
  12. [CS]
    J. Chima and E. Shult, Regular thin near polygons (to appear).Google Scholar
  13. [Co]
    B.N. Cooperstein, A finite flag-transitive geometry of extended G2 type (to appear).Google Scholar
  14. [Cu]
    C.W. Curtis, Central extensions of groups of Lie type. J. reine angew. Math. 22 (1965), 174–185.MathSciNetGoogle Scholar
  15. [CKS]
    ___, W.M. Kantor and G.M. Seitz, The 2-transitive permutation representations of the finite Chevalley groups. TAMS 218 (1976), 1–59.zbMATHMathSciNetGoogle Scholar
  16. [De]
    P. Dembowski, Finite geometries. Springer, Berlin-Heidelberg-New York 1968.zbMATHCrossRefGoogle Scholar
  17. [FH]
    W. Feit and G. Higman, The nonexistence of certain generalized polygons. J. Alg. 1 (1964), 114–131.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [Hae]
    W. Haemers, Eigenvalue techniques in design and graph theory. Ph.d. Thesis, Eindhoven 1979.Google Scholar
  19. [HaR]
    ___ and C. Roos, An inequality for generalized hexagons. Geom. Ded. 10 (1981), 219–222.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [Hal]
    M. Hall, Jr., Affine generalized quadrilaterals, in "Studies in Pure Mathematics", pp. 113–116, Academic Press 1971.Google Scholar
  21. [Hi]
    D.G. Higman, Invariant relations, coherent configurations and generalized polygons, pp. 247–363 in "Combinatorics", Reidel, Dordrecht 1975.Google Scholar
  22. [Hir]
    J.W.P. Hirschfeld, Projective geometries over finite fields. Clarendon Press, Oxford 1979.zbMATHGoogle Scholar
  23. [Ka 1]
    W.M. Kantor, Moore geometries and rank 3 groups with μ=1. Quart. J. Math. 28 (1977), 309–328.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Ka 2]
    ___, Generalized quadrangles associated with G2(q), JCT(A) 29 (1980), 212–219.zbMATHMathSciNetGoogle Scholar
  25. [Ka 3]
    ___, Some geometries that are almost buildings. Europ. J. Combinatorics 2 (1981), 239–247.zbMATHMathSciNetGoogle Scholar
  26. [Ka 4]
    ___, Spreads, translation planes and Kerdock sets. II. SIAM J. Algebraic and Discrete Methods 3 (1982), 308–318.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [Ka 5]
    ___, Translation planes of order q6 admitting SL(2,q2). JCT(A) 32 (1982), 299–302.zbMATHMathSciNetGoogle Scholar
  28. [Ka 6]
    ___, Some exceptional 2-adic buildings. J. Alg. 92 (1985), 208–223.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [Ka 7]
    ___, Some locally finite flag-transitive buildings (to appear in Europ. J. Combinatorics).Google Scholar
  30. [Ka 8]
    ___, Primitive permutation groups of odd degree, and an application to finite projective planes (to appear in J. Alg.).Google Scholar
  31. [KL]
    ___ and R.A. Liebler, The rank 3 permutation representations of the finite classical groups. TAMS 271 (1982), 1–71.zbMATHMathSciNetGoogle Scholar
  32. [KMW 1]
    P. Köhler, T. Meixner, and M. Wester, Triangle groups. Comm. in Alg. 12 (1984), 1595–1626.zbMATHCrossRefGoogle Scholar
  33. [KMW 2]
    ___, The 2-adic affine building of type Ã2 and its finite projections. JCT(A) 38 (1985), 203–209.zbMATHGoogle Scholar
  34. [KMW 3]
    ___, The affine building of type Ã2 over a local field of characteristic two. Arch. Math. 42 (1984), 400–407.zbMATHCrossRefGoogle Scholar
  35. [KMW 4]
    ___, Unpublished.Google Scholar
  36. [KS]
    R. Kilmoyer and L. Solomon, On the theorem of Feit-Higman. JCT(A) 15 (1973), 310–322.zbMATHMathSciNetGoogle Scholar
  37. [LaS]
    V. Landazuri and G.M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. J. Alg. 32 (1974), 418–433.zbMATHCrossRefMathSciNetGoogle Scholar
  38. [Li]
    Hui-Ling Li, Two pairs of simply connected geometries (to appear).Google Scholar
  39. [Lie]
    R.A. Liebler, Tactical configurations and their generic ring. (to appear).Google Scholar
  40. [MT]
    T. Meixner and F. Timmesfled, Chamber systems with string diagrams. Geom. Ded. 15 (1983), 115–123.zbMATHCrossRefGoogle Scholar
  41. [Ne]
    A. Neumaier, Some sporadic geometries related to PG(3,2). Arch. Math. 42 (1984), 89–96.zbMATHCrossRefMathSciNetGoogle Scholar
  42. [Ni]
    R. Niles, BN-pairs and finite groups with parabolic-type subgroups, J. Alg. 74 (1982), 484–494.CrossRefMathSciNetGoogle Scholar
  43. [Ot 1]
    U. Ott, Bericht über Hecke Algebren und Coxeter Algebren endlicher Geometrien, pp. 260–271 in "Finite geometries and designs", London Math. Soc. Lecture Note Series 49, 1981.Google Scholar
  44. [Ot 2]
    U. Ott, On finite geometries of type B3. JCT(A) 39 (1985), 209–221.zbMATHMathSciNetGoogle Scholar
  45. [Pa 1]
    S.E. Payne, Nonisomorphic generalized quadrangles. J. Alg. 18 (1971), 201–212.zbMATHCrossRefGoogle Scholar
  46. [PA 2]
    ___, Hyperovals yield many GQ, (to appear).Google Scholar
  47. [PaT]
    ___ and J.A. Thas, Finite generalized quadrangles. Pitman 1984.Google Scholar
  48. [Pr]
    G. Prasad, Unpublished.Google Scholar
  49. [Re]
    S. Rees, C3 geometries arising from the Klein quadric. Geom. Ded. 18 (1985), 67–85.zbMATHCrossRefMathSciNetGoogle Scholar
  50. [Ro 1]
    M.A. Ronan, A geometric characterization of Moufang hexagons. Invent. Math. 57 (1980), 227–262.zbMATHCrossRefMathSciNetGoogle Scholar
  51. [Ro 2]
    ___, Coverings and automorphisms of chamber systems. Europ. J. Combinatorics 1 (1980), 259–269.zbMATHMathSciNetGoogle Scholar
  52. [Ro 3]
    ___, Triangle geometries. JCT(A) 37 (1984), 294–319.zbMATHMathSciNetGoogle Scholar
  53. [Ro 4]
    ___, Unpublished.Google Scholar
  54. [RSm]
    ___ and S.D. Smith, 2-local geometries for some sporadic groups. Proc. Symp. Pure Math. 37 (1980), 238–289.MathSciNetGoogle Scholar
  55. [RSt]
    ___ and G. Stroth, Minimal parabolic geometries for the sporadic groups. Europ. J. Combinatorics 5 (1984), 59–92.zbMATHMathSciNetGoogle Scholar
  56. [Se]
    G.M. Seitz, Flag-transitive subgroups of Chevalley groups. Ann. Math. 97 (1973), 27–56; correction (unpublished).zbMATHCrossRefMathSciNetGoogle Scholar
  57. [St 1]
    G. Stroth, Finite Tits geometries over GF(2) related to the group A7 (to appear).Google Scholar
  58. [St 2]
    ___, Parabolic systems over GF(2) whose diagrams contain double bonds, Part I: the symplectic case (to appear).Google Scholar
  59. [Tim 1]
    F.G. Timmesfeld, Tits geometries and parabolic systems in finitely generated groups, I, II. Math. Z. 184 (1983), 337–396, 449–487.Google Scholar
  60. [Tim 2]
    ___, Tits geometries and parabolic systems of rank 3. (to appear).Google Scholar
  61. [Ti 1]
    J. Tits, Sur la trialité et certains groupes qui s'en déduisent, Publ. Math. I.H.E.S. 2 (1959), 14–60.Google Scholar
  62. [Ti 2]
    ___, Buildings of spherical type and finite BN-pairs. Springer Lecture Notes 286, 1974.Google Scholar
  63. [Ti 3]
    ___, Classification of buildings of spherical type and Moufang polygons: A survey, pp. 229–256 in "Atti Coll. Int. Teorie Combinatorie", Accad. Lincei, Rome 1976.Google Scholar
  64. [Ti 4]
    ___, Endliche, Spiegelungsgruppen, die als Weylgruppean auftreten. Invent. Math. 43 (1977), 283–295.zbMATHCrossRefMathSciNetGoogle Scholar
  65. [Ti 5]
    ___, Reductive groups over local fields. Proc. Symp. Pure Math. 33 (1979), 29–69.CrossRefMathSciNetGoogle Scholar
  66. [Ti 6]
    ___, Buildings and Buekenhout geometries, pp. 309–320 in "Finite simple groups", II", Academic Press 1980.Google Scholar
  67. [Ti 7]
    ___, A local approach to buildings, pp. 519–547 in "The Geometric Vein. The Coxeter Festschrift", Springer, New York-Heidelberg-Berlin, 1981.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • William M. Kantor
    • 1
  1. 1.University of OregonUSA

Personalised recommendations