Geometriae Dedicata

, Volume 49, Issue 2, pp 143–154 | Cite as

Slanted symplectic quadrangles

  • T. Grundhöfer
  • M. Joswig
  • M. Stroppel
Article

Abstract

By ‘slanting’ symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bichara, A., ‘Characterization of the generalized quadrangles embedded inA 3,q’,Riv. Mat. Univ. Parma (4)4 (1978), 277–290.Google Scholar
  2. 2.
    Bichara, A., Mazzocca, F. and Somma, C., ‘On the classification of generalized quadrangles in finite affine spaces AG(3,2h)’,Boll. Un. Mat. Ital. (5)17-B (1980), 298–307.Google Scholar
  3. 3.
    Coxeter, H. S. M., ‘The evolution of Coxeter-Dynkin diagrams’,Nieuw Arch. Wisk. (4)9 (1991), 233–248.Google Scholar
  4. 4.
    De Soete, M. and Thas, J. A., ‘Characterizations of the generalized quadrangles T*2(O) and T2(O)’,Ars Combin. 22 (1986), 171–186.Google Scholar
  5. 5.
    De Soete, M. and Thas, J. A., ‘ℛ-Regularity and characterizations of the generalized quadrangles P(W(s), (∞))’,Proc. Bari 1984 (eds A. Barlottiet al.),Ann. Discrete Math. 30 (1986), 171–184.Google Scholar
  6. 6.
    Dieudonné, J.,La géométrie des groupes classiques, 3rd edn, Springer, Berlin, 1971.Google Scholar
  7. 7.
    Frame, J. S., ‘A symmetric representation of the twenty-seven lines on a cubic surface by lines in a finite geometry’,Bull. Amer. Math. Soc. 44 (1938), 658–661.Google Scholar
  8. 8.
    Freudenthal, H., ‘Une étude de quelques quadrangles généralisés’,Ann. Mat. Pura Appl. (4)102 (1975), 109–133.Google Scholar
  9. 9.
    Gorenstein, D.,Finite Groups, Harper & Row, New York, 1968.Google Scholar
  10. 10.
    Grundhöfer, T. and Knarr, N., ‘Topology in generalized quadrangles’,Topology Appl. 34 (1990), 139–152.Google Scholar
  11. 11.
    Hartshorne, R.,Algebraic Geometry, Springer, New York, 1977.Google Scholar
  12. 12.
    Knarr, N., ‘Projectivities in generalized polygons’,Ars Combin. 25B (1988), 265–275.Google Scholar
  13. 13.
    Miller, G. A., Blichfeld, H. F. and Dickson, L. E.,Theory and Applications of Finite Groups, Stechert, New York, 1938.Google Scholar
  14. 14.
    Mumford, D.,Algebraic Geometry I, Complex Projective Varieties, Springer, Berlin, 1976.Google Scholar
  15. 15.
    Payne, S. E., ‘Nonisomorphic generalized quadrangles’,J. Algebra 18 (1971), 201–212.Google Scholar
  16. 16.
    Payne, S. E., ‘The equivalence of certain generalized quadrangles’,J. Combin. Theory Ser. A 10 (1971), 284–289.Google Scholar
  17. 17.
    Payne, S. E., ‘The generalized quadrangle with (s,t)=(3,5)’,Proc. Boca Raton 1990,Congr. Numerantium 77 (1990), 5–29.Google Scholar
  18. 18.
    Payne, S. E. and Thas, J. A.,Finite Generalized Quadrangles, Pitman, London, 1984.Google Scholar
  19. 19.
    Somma, C., ‘Generalized quadrangles with parallelism’,Proc. Trento 1980 (ed. A. Barlotti),Annals Discrete Math. 14 (1982), 265–282.Google Scholar
  20. 20.
    Stroppel, M., ‘Reconstruction of incidence structures from groups of automorphisms’,Arch. Math. (Basel)58 (1992), 621–624.Google Scholar
  21. 21.
    Stroppel, M., ‘A categorical glimpse at the reconstruction of geometries’,Geom. Dedicata (to appear).Google Scholar
  22. 22.
    Tamaschke, O.,Projektive Geometrie, II, Bibliogr. Inst., Mannheim, 1972.Google Scholar
  23. 23.
    Szambien, H., ‘Minimal topological projective planes’,J. Geometry 35 (1989), 177–185.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • T. Grundhöfer
    • 1
  • M. Joswig
    • 1
  • M. Stroppel
    • 2
  1. 1.Mathematisches InstitutUniversität TübingenTübingenGermany
  2. 2.Fachbereich MathematikTechnische Hochschule DarmstadtDarmstadtGermany

Personalised recommendations