Ebook: Symmetry in Finite Generalized Quadrangles

In this monograph finite generalized quadrangles are classified by symmetry, generalizing the celebrated Lenz-Barlotti classification for projective planes. The book is self-contained and serves as introduction to the 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.

---

In this monograph finite generalized quadrangles are classified by symmetry, generalizing the celebrated Lenz-Barlotti classification for projective planes. The book is self-contained and serves as introduction to the 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.

---

In this monograph finite generalized quadrangles are classified by symmetry, generalizing the celebrated Lenz-Barlotti classification for projective planes. The book is self-contained and serves as introduction to the 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.

Content:
Finite Generalized Quadrangles ....Pages 1-14
Elation Generalized Quadrangles, Translation Generalized Quadrangles and Flocks ....Pages 15-30
The Known Generalized Quadrangles ....Pages 31-42
Substructures of Finite Nets ....Pages 43-50
Symmetry-Class I: Generalized Quadrangles without Axes of Symmetry ....Pages 51-56
Symmetry-Class II: Concurrent Axes of Symmetry in Generalized Quadrangles ....Pages 57-84
Symmetry-Class ≥ III: Span-Symmetric Generalized Quadrangles ....Pages 85-108
Generalized Quadrangles with Distinct Translation Points ....Pages 109-114
The Classification Theorem ....Pages 115-121
Symmetry-Class IV.3: Translation Duals of TGQ’s which Arise from Flocks ....Pages 123-136
A Characterization Theorem and a Classification Theorem ....Pages 137-142
The Symmetry-Class V ....Pages 143-154
Recapitulation of the Classification Theorem ....Pages 155-160
Blueprint for the Classification of Translation Generalized Quadrangles ....Pages 161-170
Appendix A. Semi Quadrangles ....Pages 171-188
Appendix B. The Sizes of Some Groups and a Table of Schur Multipliers ....Pages 189-192