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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bichara, A., ‘Characterization of the generalized quadrangles embedded inA 3,q’,Riv. Mat. Univ. Parma (4)4 (1978), 277–290.
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.
Coxeter, H. S. M., ‘The evolution of Coxeter-Dynkin diagrams’,Nieuw Arch. Wisk. (4)9 (1991), 233–248.
De Soete, M. and Thas, J. A., ‘Characterizations of the generalized quadrangles T*2(O) and T2(O)’,Ars Combin. 22 (1986), 171–186.
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.
Dieudonné, J.,La géométrie des groupes classiques, 3rd edn, Springer, Berlin, 1971.
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.
Freudenthal, H., ‘Une étude de quelques quadrangles généralisés’,Ann. Mat. Pura Appl. (4)102 (1975), 109–133.
Gorenstein, D.,Finite Groups, Harper & Row, New York, 1968.
Grundhöfer, T. and Knarr, N., ‘Topology in generalized quadrangles’,Topology Appl. 34 (1990), 139–152.
Hartshorne, R.,Algebraic Geometry, Springer, New York, 1977.
Knarr, N., ‘Projectivities in generalized polygons’,Ars Combin. 25B (1988), 265–275.
Miller, G. A., Blichfeld, H. F. and Dickson, L. E.,Theory and Applications of Finite Groups, Stechert, New York, 1938.
Mumford, D.,Algebraic Geometry I, Complex Projective Varieties, Springer, Berlin, 1976.
Payne, S. E., ‘Nonisomorphic generalized quadrangles’,J. Algebra 18 (1971), 201–212.
Payne, S. E., ‘The equivalence of certain generalized quadrangles’,J. Combin. Theory Ser. A 10 (1971), 284–289.
Payne, S. E., ‘The generalized quadrangle with (s,t)=(3,5)’,Proc. Boca Raton 1990,Congr. Numerantium 77 (1990), 5–29.
Payne, S. E. and Thas, J. A.,Finite Generalized Quadrangles, Pitman, London, 1984.
Somma, C., ‘Generalized quadrangles with parallelism’,Proc. Trento 1980 (ed. A. Barlotti),Annals Discrete Math. 14 (1982), 265–282.
Stroppel, M., ‘Reconstruction of incidence structures from groups of automorphisms’,Arch. Math. (Basel)58 (1992), 621–624.
Stroppel, M., ‘A categorical glimpse at the reconstruction of geometries’,Geom. Dedicata (to appear).
Tamaschke, O.,Projektive Geometrie, II, Bibliogr. Inst., Mannheim, 1972.
Szambien, H., ‘Minimal topological projective planes’,J. Geometry 35 (1989), 177–185.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grundhöfer, T., Joswig, M. & Stroppel, M. Slanted symplectic quadrangles. Geom Dedicata 49, 143–154 (1994). https://doi.org/10.1007/BF01610617
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01610617