| Preliminaries Introduction Finite Generalized Quadrangles Prolegomena 1 q-Clans and Their Geometries 1.1 Anisotropism 1.2 q-Clans 1.3 Flocks of a Quadratic Cone 1.4 4-Gonal Families from q-Clans 1.5 Ovals in Ra 1.6 Herd Cover and Herd of Ovals 1.7 Herds of Ovals from q-Clans 1.8 Generalized Quadrangles from q-Clans 1.9 Spreads of PG(3,q) Associated with q-Clans 2 The Fundamental Theorem 2.1 Grids and Affine Planes 2.2 The Fundamental Theorem 2.3 Aut(G) 2.4 Extension to 1/2-Normalized q-Clans 2.5 A Characterization of the q-Clan Kernel 2.6 Very Important Concept 2.7 The q-clan Cis, s F 2.8 The Induced Oval Stabilizers 2.9 Action of H on Generators of Cone K 3 Aut(GQ(C)) 3.1 General Remarks 3.2 An Involution of GQ(C) 3.3 The Automorphism Group of the Herd Cover 3.4 The Magic Action of O'Keefe and Penttila 3.5 The Automorphism Group of the Herd 3.6 The Groups Go, Go and G0 3.7 The Square-Bracket Function 3.8 A Cyclic Linear Collineation 3.9 Some Involutions 3.10 Some Semi-linear Collineations 4 The Cyclic q-Clans 4.1 The Unified Construction of [COP03] 4.2 The Known Cyclic q-Clans 4.3 q-Clan Functions Via the Square Bracket 4.4 The Flip is a Collineation 4.5 The Main Isomorphism Theorem 4.6 The Unified Construction Gives Cyclic q-Clans 4.7 Some Semi-linear Collineations 4.8 An Oval Stabilizer 5 Applications to the Known Cyclic q-Clans 5.1 The Classical Examples: q=2e for e ≥1 5.2 The FTWKB Examples: q=2e with e Odd 5.3 The Subiaco Examples: q=2e, e≥4 5.4 The Adelaide Examples: q=2e with e Even 6 The Subiaco Oval Stabilizers 6.1 Algebraic Plane Curves 6.2 The Action of Go on the Ra 6.3 The case e≡2 (mod 4) 6.4 The Case e≡10 (rood 20) 6.5 Subiaco Hyperovals: The Various Cases 6.6 O+(1,1) as an Algebraic Curve 6.7 The Case e≡O (rood 4) 6.8 The Case e Odd 6.9 The case e≡2 (rood 4) 6.10 Summary of Subiaco Oval Stabilizers 7 The Adelaide Oval Stabilizers 7.1 The Adelaide Oval 7.2 A Polynomial Equation for the Adelaide Oval 7.3 Irreducibility of the Curve 7.4 The Complete Oval Stabilizer 8 The Payne q-Clans 9 Other Good Stuff Bibliography Index |
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