Symmetry in Finite Generalized Quadrangles

In this self-contained monograph, finite generalized quadrangles are classified by symmetry, generalizing the celebrated Lenz--Barlotti classification for projective planes. It serves as an introduction to the combinatorial, geometrical, and group-theoretical concepts that arise in the classification and the general theory of finite generalized quadrangles. Several long-standing open problems are solved by using a mixture of geometrical, combinatorial, and group-theoretical arguments. Further, a conjectural classification program for all finite transaltion generalized quadrangles is decribed. The book will be useful to advanced graduate students and researchers in geometry, group theory, or algebraic combinatorics.

Contains many new results, not previously published anywhere else Proof of the main (classification) result is written and split up in such a way that many parts of it can be seen in their own right, and can be used independently Several open problems and longstanding conjectures are completely solved in the book, often by introducing new techniques which can be used in various other situations

Klappentext

This monograph classifies finite generalized quadrangles by symmetry, generalizing the celebrated Lenz-Barlotti classification for projective planes. The book introduces combinatorial, geometrical and group-theoretical concepts that arise in the classification and in the general theory of finite generalized quadrangles, including automorphism groups, elation and translation generalized quadrangles, generalized ovals and generalized ovoids, span-symmetric generalized quadrangles, flock geometry and property (G), regularity and nets, split BN-pairs of rank 1, and the Moufang property.

Inhalt
Introduction: History, Motivation.- 1. Finite Generalized Quadrangles.- 2. Elation Generalized Quadrangles, Translation Generalized Quadrangles and Flocks.- 3. The Known Generalized Quadrangles.- 4. Substructures of Finite Nets.- 5. Symmetry Class I: Generalized Quadrangles with Axes of Symmetry.- 6. Symmetry Class II: Concurrent Axes of Symmetry in Generalized Quadrangles.- 7. Symmetry Class II: Span-Symmetric Generalized Quadrangles.- 8. Generalized Quadrangles with Distinct Translation Points.- 9. The Classification Theorem.- 10. Symmetry Class IV.3: TGQs which Arise from Flocks .- 11. A Characterization Theorem and a Classification Theorem.- 12. Symmetry Class V.- 13. Recapitulation of the Classification Theorem.- 14. Semi Quadrangles.- Appendices.- References.