Finite generalized quadrangles

Preface......Page 6
Preface to the first edition......Page 8
Axioms and definitions......Page 14
Restrictions on the parameters......Page 15
Regularity, antiregularity, semiregularity, and property (H)......Page 17
An application of the Higman–Sims technique......Page 20
Regularity revisited......Page 23
Semiregularity and property (H)......Page 24
Triads with exactly 1+t/s centers......Page 27
Ovoids, spreads and polarities......Page 29
Automorphisms......Page 32
A second application of Higman–Sims......Page 33
The basic inequalities......Page 35
Recognizing subquadrangles......Page 36
Automorphisms and subquadrangles......Page 37
Semiregularity, property (H) and subquadrangles......Page 38
3-Regularity and subquadrangles......Page 40
k-arcs and subquadrangles......Page 41
Description of the known GQ......Page 44
Isomorphisms between the known GQ......Page 50
Combinatorial properties: regularity, antiregularity, semiregularity, property (H)......Page 55
Ovoids and spreads of the known GQ......Page 59
Subquadrangles......Page 62
Symmetric designs derived from GQ......Page 63
The tangent hyperplane......Page 68
Embedding ==========S in a polarity: preliminary results......Page 71
The finite case......Page 73
Characterizations of W(q) and Q(4,q)......Page 77
Characterizations of T_3(O) and Q(5,q)......Page 81
Tallini's characterization of H(3,s)......Page 95
Characterizations of H(4,q2)......Page 102
Additional characterizations......Page 107
s=2......Page 114
s=3......Page 115
s=4......Page 120
Embedding in AG(3,s+1)......Page 139
Embedding in AG(4,s+1)......Page 144
Embedding in AG(d,s+1), d5......Page 148
Whorls, elations and symmetries......Page 151
Elation generalized quadrangles......Page 152
Recognizing TGQ......Page 156
Fixed substructures of translations......Page 157
The kernel of a TGQ......Page 159
The structure of TGQ......Page 160
T(n,m,q)......Page 161
Definitions and an easy theorem......Page 167
The Moufang conditions and property (H)......Page 168
Moufang conditions and TGQ......Page 169
An application of the Higman–Sims technique......Page 172
The case (iv) of 9.2.5......Page 174
The extremal case q = 1 + s2 +......Page 176
A theorem of M. A. Ronan......Page 180
Other classifications using collineations......Page 183
4-gonal families......Page 184
4-gonal partitions......Page 185
Explicit description of 4-gonal families for TGQ......Page 186
A model for STGQ......Page 187
A model for certain STGQ with (s,t)=(q2,q)......Page 190
Examples of STGQ with order (q2,q)......Page 195
4-gonal bases: span-symmetric GQ......Page 201
One axis of symmetry......Page 207
Two concurrent axes of symmetry......Page 212
Three concurrent axes of symmetry......Page 216
The kernel of a T-set up......Page 222
Admissible pairs......Page 226
Admissible pairs of additive permutations......Page 229
Collineations......Page 235
Generalized quadrangles T_2(O)......Page 238
Isomorphisms......Page 244
Nonisomorphic GQ......Page 247
The ovals of D. Glynn......Page 249
Partial geometries, partial quadrangles and semi partial geometries......Page 254
Partial 3-spaces......Page 255
Partial geometric designs......Page 256
Generalized polygons......Page 257
Polar spaces and Shult spaces......Page 258
Pseudo-geometric and geometric graphs......Page 259
Bibliography......Page 260
GQ, q-clans, flocks, planes, property (G) and hyperovals......Page 272
q-clan geometry......Page 276
Translation generalized quadrangles......Page 280
Subquadrangles, ovoids and spreads......Page 285
Automorphisms of generalized quadrangles......Page 287
Embeddings, tight subsets, extensions......Page 289
Bibliography......Page 292
Index......Page 298