This is my favorite finite geometry text. The treatment is a mixture of basic and complex and is hence suitable for a wise variety of readers, probably best for undergraduate/beginning graduate courses, but works well for self-study. I am excited about the generalized quadrangles sections.
Author(s): Lynn Margaret Batten
Edition: 2
Publisher: Cambridge University Press
Year: 1997
Language: English
Pages: 207
Contents ......Page 7
Preface ......Page 11
Preface to the first edition ......Page 13
1.1 Some basic concepts: consistency and dependence ......Page 15
1.2 Near-linear spaces ......Page 18
1.3 New near-linear spaces from old ......Page 20
1.4 Dimension ......Page 23
1.5 Incidence matrices ......Page 25
1.6 The connection number ......Page 28
1.7 Linear functions ......Page 31
1.8 Exercises ......Page 33
2.1 Examples ......Page 37
2.2 The de Bruijn-Erdos theorem ......Page 39
2.3 Numerical properties ......Page 41
2.4 The exchange property ......Page 44
2.5 Hyperplanes ......Page 46
2.6 Linear functions ......Page 48
2.7 Exercises ......Page 51
3.1 Projective planes ......Page 55
3.2 Finite projective planes ......Page 57
3.3 Embedding a near-linear space in a projective plane ......Page 58
3.4 Subplanes ......Page 59
3.5 Collineations in projective planes ......Page 60
3.6 The Desargues configuration ......Page 63
3.7 Construction of projective planes from vector spaces ......Page 65
3.8 The Pappus configuration ......Page 71
3.9 Projective spaces ......Page 74
3.10 Desargues configurations again ......Page 77
3.11 Exercises ......Page 78
4.1 Affine planes ......Page 81
4.2 Finite affine planes ......Page 83
4.3 Embedding an affine plane in a projective plane ......Page 84
4.4 Collineations in affine planes ......Page 85
4.5 The Desargues configuration in affine planes ......Page 91
4.6 Co-ordinatization in affine planes and the Pappus configuration ......Page 93
4.7 Affine spaces ......Page 96
4.8 Exercises ......Page 99
5.1 The definition ......Page 103
5.2 Absolute points ......Page 105
5.3 Quadrics ......Page 108
5.4 Linear subspaces ......Page 113
5.5 Irreducibility ......Page 117
5.6 Projective spaces inside polar spaces ......Page 119
5.7 A history of polar spaces ......Page 121
5.8 Exercises ......Page 123
6.1 Definition and some basic results ......Page 126
6.2 All known examples ......Page 132
6.3 Some combinatorial properties ......Page 134
6.4 Generalized quadrangles with s = I = 3 ......Page 138
6.5 Subquadrangles ......Page 141
6.6 Collineations of generalized quadrangles ......Page 145
6.7 A brief history of generalized quadrangles ......Page 148
6.8 Exercises ......Page 149
7.1 The definition ......Page 152
7.2 A method of constructing proper partial geometries ......Page 155
7.3 Strongly regular graphs ......Page 156
7.4 Subgeometries ......Page 161
7.5 Pasch's axiom ......Page 163
7.6 A history of partial geometries ......Page 168
7.7 Exercises ......Page 169
8.1 Definition and examples ......Page 172
8.2 Blocking sets in projective planes ......Page 173
8.3 Blocking sets in affine planes ......Page 177
8.4 Blocking sets in Steiner systems ......Page 179
8.5 Blocking sets in generalized quadrangles and partial geometries ......Page 181
8.6 Applications of blocking sets ......Page 185
8.7 Exercises ......Page 187
Bibliography ......Page 190
Index of notation ......Page 204
Subject index ......Page 205
