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Generalized polygons, SCABs and GABs

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Buildings and the Geometry of Diagrams

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1181))

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References

  1. M. Aschbacher, Flag structures on Tits geometries. Geom. Ded. 14 (1983), 21–32.

    Article  MATH  MathSciNet  Google Scholar 

  2. ___, Finite geometries of type C3 with flag transitive groups. Geom. Ded. 16 (1984), 195–200.

    Article  MATH  MathSciNet  Google Scholar 

  3. ___ and S.D. Smith, Tits geometries over GF(2) defined by groups over GF(3). Comm. in Alg. 11 (1983), 1675–1684.

    Article  MathSciNet  Google Scholar 

  4. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. A.E. Brouwver, A.M. Cohen and J. Tits, Some remarks on Tits geometries. Indag. Math. 45 (1983), 393–400.

    Google Scholar 

  6. N.L. Biggs, Algebraic graph theory. Cambridge University Press, Cambridge 1974.

    Book  MATH  Google Scholar 

  7. 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.

    MATH  MathSciNet  Google Scholar 

  8. F. Buekenhout, Diagrams for geometries and groups. JCT(A) 27 (1979), 121–151.

    MATH  MathSciNet  Google Scholar 

  9. P.J. Cameron, Partial quadrangles. Quart. J. Math. 26 (1974), 61–73.

    Article  Google Scholar 

  10. ___ and W.M. Kantor, 2-Transitive and antiflag transitive collineation groups of finite projective and polar spaces. J. Alg. 60 (1979), 384–422.

    Article  MATH  MathSciNet  Google Scholar 

  11. R.W. Carter, Simple groups of Lie type. Wiley, New York 1972.

    MATH  Google Scholar 

  12. J. Chima and E. Shult, Regular thin near polygons (to appear).

    Google Scholar 

  13. B.N. Cooperstein, A finite flag-transitive geometry of extended G2 type (to appear).

    Google Scholar 

  14. C.W. Curtis, Central extensions of groups of Lie type. J. reine angew. Math. 22 (1965), 174–185.

    MathSciNet  Google Scholar 

  15. ___, W.M. Kantor and G.M. Seitz, The 2-transitive permutation representations of the finite Chevalley groups. TAMS 218 (1976), 1–59.

    MATH  MathSciNet  Google Scholar 

  16. P. Dembowski, Finite geometries. Springer, Berlin-Heidelberg-New York 1968.

    Book  MATH  Google Scholar 

  17. W. Feit and G. Higman, The nonexistence of certain generalized polygons. J. Alg. 1 (1964), 114–131.

    Article  MATH  MathSciNet  Google Scholar 

  18. W. Haemers, Eigenvalue techniques in design and graph theory. Ph.d. Thesis, Eindhoven 1979.

    Google Scholar 

  19. ___ and C. Roos, An inequality for generalized hexagons. Geom. Ded. 10 (1981), 219–222.

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Hall, Jr., Affine generalized quadrilaterals, in "Studies in Pure Mathematics", pp. 113–116, Academic Press 1971.

    Google Scholar 

  21. D.G. Higman, Invariant relations, coherent configurations and generalized polygons, pp. 247–363 in "Combinatorics", Reidel, Dordrecht 1975.

    Google Scholar 

  22. J.W.P. Hirschfeld, Projective geometries over finite fields. Clarendon Press, Oxford 1979.

    MATH  Google Scholar 

  23. W.M. Kantor, Moore geometries and rank 3 groups with μ=1. Quart. J. Math. 28 (1977), 309–328.

    Article  MATH  MathSciNet  Google Scholar 

  24. ___, Generalized quadrangles associated with G2(q), JCT(A) 29 (1980), 212–219.

    MATH  MathSciNet  Google Scholar 

  25. ___, Some geometries that are almost buildings. Europ. J. Combinatorics 2 (1981), 239–247.

    MATH  MathSciNet  Google Scholar 

  26. ___, Spreads, translation planes and Kerdock sets. II. SIAM J. Algebraic and Discrete Methods 3 (1982), 308–318.

    Article  MATH  MathSciNet  Google Scholar 

  27. ___, Translation planes of order q6 admitting SL(2,q2). JCT(A) 32 (1982), 299–302.

    MATH  MathSciNet  Google Scholar 

  28. ___, Some exceptional 2-adic buildings. J. Alg. 92 (1985), 208–223.

    Article  MATH  MathSciNet  Google Scholar 

  29. ___, Some locally finite flag-transitive buildings (to appear in Europ. J. Combinatorics).

    Google Scholar 

  30. ___, Primitive permutation groups of odd degree, and an application to finite projective planes (to appear in J. Alg.).

    Google Scholar 

  31. ___ and R.A. Liebler, The rank 3 permutation representations of the finite classical groups. TAMS 271 (1982), 1–71.

    MATH  MathSciNet  Google Scholar 

  32. P. Köhler, T. Meixner, and M. Wester, Triangle groups. Comm. in Alg. 12 (1984), 1595–1626.

    Article  MATH  Google Scholar 

  33. ___, The 2-adic affine building of type Ã2 and its finite projections. JCT(A) 38 (1985), 203–209.

    MATH  Google Scholar 

  34. ___, The affine building of type Ã2 over a local field of characteristic two. Arch. Math. 42 (1984), 400–407.

    Article  MATH  Google Scholar 

  35. ___, Unpublished.

    Google Scholar 

  36. R. Kilmoyer and L. Solomon, On the theorem of Feit-Higman. JCT(A) 15 (1973), 310–322.

    MATH  MathSciNet  Google Scholar 

  37. V. Landazuri and G.M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. J. Alg. 32 (1974), 418–433.

    Article  MATH  MathSciNet  Google Scholar 

  38. Hui-Ling Li, Two pairs of simply connected geometries (to appear).

    Google Scholar 

  39. R.A. Liebler, Tactical configurations and their generic ring. (to appear).

    Google Scholar 

  40. T. Meixner and F. Timmesfled, Chamber systems with string diagrams. Geom. Ded. 15 (1983), 115–123.

    Article  MATH  Google Scholar 

  41. A. Neumaier, Some sporadic geometries related to PG(3,2). Arch. Math. 42 (1984), 89–96.

    Article  MATH  MathSciNet  Google Scholar 

  42. R. Niles, BN-pairs and finite groups with parabolic-type subgroups, J. Alg. 74 (1982), 484–494.

    Article  MathSciNet  Google Scholar 

  43. 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. U. Ott, On finite geometries of type B3. JCT(A) 39 (1985), 209–221.

    MATH  MathSciNet  Google Scholar 

  45. S.E. Payne, Nonisomorphic generalized quadrangles. J. Alg. 18 (1971), 201–212.

    Article  MATH  Google Scholar 

  46. ___, Hyperovals yield many GQ, (to appear).

    Google Scholar 

  47. ___ and J.A. Thas, Finite generalized quadrangles. Pitman 1984.

    Google Scholar 

  48. G. Prasad, Unpublished.

    Google Scholar 

  49. S. Rees, C3 geometries arising from the Klein quadric. Geom. Ded. 18 (1985), 67–85.

    Article  MATH  MathSciNet  Google Scholar 

  50. M.A. Ronan, A geometric characterization of Moufang hexagons. Invent. Math. 57 (1980), 227–262.

    Article  MATH  MathSciNet  Google Scholar 

  51. ___, Coverings and automorphisms of chamber systems. Europ. J. Combinatorics 1 (1980), 259–269.

    MATH  MathSciNet  Google Scholar 

  52. ___, Triangle geometries. JCT(A) 37 (1984), 294–319.

    MATH  MathSciNet  Google Scholar 

  53. ___, Unpublished.

    Google Scholar 

  54. ___ and S.D. Smith, 2-local geometries for some sporadic groups. Proc. Symp. Pure Math. 37 (1980), 238–289.

    MathSciNet  Google Scholar 

  55. ___ and G. Stroth, Minimal parabolic geometries for the sporadic groups. Europ. J. Combinatorics 5 (1984), 59–92.

    MATH  MathSciNet  Google Scholar 

  56. G.M. Seitz, Flag-transitive subgroups of Chevalley groups. Ann. Math. 97 (1973), 27–56; correction (unpublished).

    Article  MATH  MathSciNet  Google Scholar 

  57. G. Stroth, Finite Tits geometries over GF(2) related to the group A7 (to appear).

    Google Scholar 

  58. ___, Parabolic systems over GF(2) whose diagrams contain double bonds, Part I: the symplectic case (to appear).

    Google Scholar 

  59. 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. ___, Tits geometries and parabolic systems of rank 3. (to appear).

    Google Scholar 

  61. 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. ___, Buildings of spherical type and finite BN-pairs. Springer Lecture Notes 286, 1974.

    Google Scholar 

  63. ___, 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. ___, Endliche, Spiegelungsgruppen, die als Weylgruppean auftreten. Invent. Math. 43 (1977), 283–295.

    Article  MATH  MathSciNet  Google Scholar 

  65. ___, Reductive groups over local fields. Proc. Symp. Pure Math. 33 (1979), 29–69.

    Article  MathSciNet  Google Scholar 

  66. ___, Buildings and Buekenhout geometries, pp. 309–320 in "Finite simple groups", II", Academic Press 1980.

    Google Scholar 

  67. ___, A local approach to buildings, pp. 519–547 in "The Geometric Vein. The Coxeter Festschrift", Springer, New York-Heidelberg-Berlin, 1981.

    Chapter  Google Scholar 

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Authors and Affiliations

  1. University of Oregon, USA

    William M. Kantor

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  1. William M. Kantor
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Editors and Affiliations

  1. Istituto Matematico “U.DINI”, Università di Firenze, Viale Morgagni 67/a, 50134, Firenze, Italy

    Luigi A. Rosati

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© 1986 Springer-Verlag Berlin Heidelberg

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Kantor, W.M. (1986). Generalized polygons, SCABs and GABs. In: Rosati, L.A. (eds) Buildings and the Geometry of Diagrams. Lecture Notes in Mathematics, vol 1181. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075513

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  • DOI: https://doi.org/10.1007/BFb0075513

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  • Print ISBN: 978-3-540-16466-1

  • Online ISBN: 978-3-540-39801-1

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