Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T02:56:01.009Z Has data issue: false hasContentIssue false
  • English
  • Français

ELATION GENERALIZED QUADRANGLES FOR WHICH THE NUMBER OF LINES ON A POINT IS THE SUCCESSOR OF A PRIME

Part of: Finite geometry and special incidence structures

Published online by Cambridge University Press:  01 December 2008

JOHN BAMBERG ,
TIM PENTTILA  and
CSABA SCHNEIDER
JOHN BAMBERG*
Affiliation:
Department of Pure Mathematics, Ghent University, Galglaan 2, B-9000 Ghent, Belgium (email: [email protected])
TIM PENTTILA
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA (email: [email protected])
CSABA SCHNEIDER
Affiliation:
Centro de Álgebra da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that an elation generalized quadrangle that has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (that is, is a flock quadrangle).

Keywords

elation generalized quadrangleflockp-groupKantor family

MSC classification

Secondary: 51E12: Generalized quadrangles, generalized polygons
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 3 , December 2008 , pp. 289 - 303
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This research formed part of an Australian Research Council Discovery Grant project that was undertaken at the University of Western Australia. The first author was supported by a Marie Curie Incoming International Fellowship within the 6th European Community Framework Programme, contract number MIIF1-CT-2006-040360. The third author was supported by the Hungarian Scientific Research Fund (OTKA) grant F049040; he is also grateful to Hendrik Van Maldeghem for funding his visit to the workshop Groups and Buildings 2007 held in Ghent.

References

[1]Aschbacher, M., Finite Group Theory, 2nd edn, Cambridge Studies in Advanced Mathematics, 10 (Cambridge University Press, Cambridge, 2000).Google Scholar
[2]Bader, L., Lunardon, G. and Thas, J. A., ‘Derivation of flocks of quadratic cones’, Forum Math. 2(2) (1990), 163–174.CrossRefGoogle Scholar
[3]Ball, S., Govaerts, P. and Storme, L., ‘On ovoids of parabolic quadrics’, Des. Codes Cryptogr. 38(1) (2006), 131–145.Google Scholar
[4]Bloemen, I., Thas, J. A. and Van Maldeghem, H., ‘Elation generalized quadrangles of order (p,t), p prime, are classical’, J. Statist. Plann. Inference 56(1) (1996), 49–55. (Special issue on orthogonal arrays and affine designs, Part I.)Google Scholar
[5]Brouns, L., Thas, J. A. and Van Maldeghem, H., ‘A characterization of Q(5,q) using one subquadrangle Q(4,q)’, European J. Combin. 23(2) (2002), 163–177.Google Scholar
[6]Brown, M. R., ‘A characterisation of the generalized quadrangle Q(5,q) using cohomology’, J. Algebraic Combin. 15(2) (2002), 107–125.CrossRefGoogle Scholar
[7]Frohardt, D., ‘Groups which produce generalized quadrangles’, J. Combin. Theory Ser. A 48(1) (1988), 139–145.CrossRefGoogle Scholar
[8]Hachenberger, D., ‘Groups admitting a Kantor family and a factorized normal subgroup’, Des. Codes Cryptogr. 8(1–2) (1996), 135–143. (Special issue dedicated to Hanfried Lenz.)CrossRefGoogle Scholar
[9]Kantor, W. M., ‘Generalized quadrangles, flocks, and BLT sets’, J. Combin. Theory Ser. A 58(1) (1991), 153–157.Google Scholar
[10]Knarr, N., ‘A geometric construction of generalized quadrangles from polar spaces of rank three’, Results Math. 21(3–4) (1992), 332–344.Google Scholar
[11]Law, M. and Penttila, T., ‘Classification of flocks of the quadratic cone over fields of order at most 29’, Adv. Geom. (Suppl.) (2003), S232–S244 (Special issue dedicated to Adriano Barlotti.)CrossRefGoogle Scholar
[12]Payne, S. E., ‘A new infinite family of generalized quadrangles’, Congr. Numer. 49 (1985), 115–128.Google Scholar
[13]Payne, S. E., ‘An essay on skew translation generalized quadrangles’, Geom. Dedicata 32(1) (1989), 93–118.Google Scholar
[14]Payne, S. E. and Thas, J. A., Finite Generalized Quadrangles, Research Notes in Mathematics, 110 (Advanced Publishing Program) (Pitman, Boston, MA, 1984).Google Scholar
[15]Robinson, D. J. S., A Course in the Theory of Groups, 2nd edn, Graduate Texts in Mathematics, 80 (Springer, New York, 1996).Google Scholar
[16]Schneider, C., ‘Groups of prime-power order with a small second derived quotient’, J. Algebra 266(2) (2003), 539–551.Google Scholar
[17]Thas, J. A., ‘Generalized quadrangles and flocks of cones’, European J. Combin. 8(4) (1987), 441–452.Google Scholar