
Preface 
v 




Chapter 1: Groups Acting on Sets 
1 
1.1 
Exercises 
4 




Chapter 2: Affine Geometry

6 
2.1 
Semilinear transformations 
7 
2.2 
The affine group 
9 
2.3 
Exercises 
10 




Chapter 3: Projective Geometry 
13 
3.1 
Axioms for projective geometry 
15 
3.2 
Exercises 
16 




Chapter 4: The General and Special Linear Groups 
18 
4.1 
The dual space 
18 
4.2 
The groups SL(V) and PSL(V) 
19 
4.3 
Order formulae 
19 
4.4 
The action of PSL(V) on P(V) 
20 
4.5 
Transvections 
20 
4.6 
The simplicity of PSL(V) 
22 
4.7 
The groups PSL(2,q) 
23 
4.8 
Exercises 
25 




Chapter 5: BNPairs and Buildings 
27 
5.1 
The BNpair axioms 
27 
5.2 
The Tits building 
28 
5.3 
The BNpair of SL(V) 
28 
5.4 
Chambers 
30 
5.5 
Flags and apartments 
30 
5.6 
Panels 
32 
5.7 
Split BNpairs 
33 
5.8 
Commutator relations 
34 
5.9 
The Weyl group 
35 
5.10 
Exercises 
36 




Chapter 6: The 7Point Plane and the group A_{7} 
40 
6.1 
The 7point plane 
40 
6.2 
The simple group of order 168 
41 
6.3 
A geometry of 7point planes 
43 
6.4 
A geometry for A_{8} 
45 
6.5 
Exercises 
46 




Chapter 7: Polar Geometry 
50 
7.1 
The dual space 
50 
7.2 
Correlations 
51 
7.3 
Sesquilinear forms 
52 
7.4 
Polarities 
53 
7.5 
Quadratic forms 
54 
7.6 
Witt's theorem 
55 
7.7 
Bases of orthogonal hyperbolic pairs 
59 
7.8 
The group ΓL*(V) 
60 
7.9 
Flags and frames 
61 
7.10 
The building of a polarity 
63 
7.11 
Exercises 
65 




Chapter 8: Symplectic Groups 
68 
8.1 
Matrices 
68 
8.2 
Symplectic Bases 
69 
8.3 
Order formulae 
70 
8.4 
The action of PSp(V) on P(V) 
70 
8.5 
Symplectic transvections 
71 
8.6 
The simplicity of PSp(V) 
72 
8.7 
Symmetric groups 
74 
8.8 
Symplectic BNpairs 
75 
8.9 
Symplectic Buildings 
77 
8.10 
Exercises 
78 




Chapter 9: BNPairs, Diagrams and Geometries 
83 
9.1 
The BNpair of a polar building 
83 
9.2 
The Weyl group 
87 
9.3 
Coxeter groups 
90 
9.4 
The exchange condition 
91 
9.5 
Reflections and the strong exchange condition 
94 
9.6 
Parabolic subgroups of Coxeter groups 
96 
9.7 
Complexes 
97 
9.8 
Coxeter complexes 
98 
9.9 
Buildings 
99 
9.10 
Chamber systems 
101 
9.11 
Diagram geometries 
103 
9.12 
Abstract polar spaces 
107 
9.13 
Exercises 
108 




Chapter 10: Unitary Groups 
114 
10.1 
Matrices 
114 
10.2 
The field F 
115 
10.3 
Hyperbolic pairs 
116 
10.4 
Order formulae 
117 
10.5 
Unitary transvections 
118 
10.6 
Hyperbolic lines 
119 
10.7 
The action of PSU(V) on isotropic points 
120 
10.8 
Threedimensional unitary groups 
121 
10.9 
The group PSU(3,2) 
123 
10.10 
The group SU(4,2) 
125 
10.11 
The simplicity of PSU(V) 
127 
10.12 
An example 
130 
10.13 
Unitary BNpairs 
130 
10.14 
Exercises 
131 




Chapter 11: Orthogonal Groups 
136 
11.1 
Matrices 
137 
11.2 
Finite Fields 
138 
11.3 
Order formulae  one 
140 
11.4 
Threedimensional orthogonal groups 
142 
11.5 
Degenerate polar forms and the group O(2m+1,2^{k}) 
143 
11.6 
Reflections 
144 
11.7 
Root groups 
146 
11.8 
Siegel transformations 
148 
11.9 
The action of PΩ(V) on singular points 
150 
11.10 
Wall's parametrization of O(V) 
153 
11.11 
Factorization theorems 
155 
11.12 
The generation of O(V) by reflections 
156 
11.13 
Dickson's invariant 
160 
11.14 
The simplicity of PΩ(V) 
160 
11.15 
The spinor norm 
163 
11.16 
Order formulae  two 
165 
11.17 
The groups PΩ(2m+1,q), q odd 
166 
11.18 
Orthogonal BNpairs 
168 
11.19 
Maximal totally singular subspaces 
170 
11.20 
The oriflamme geometry 
172 
11.21 
Exercises 
174 




Chapter 12: The Klein Correspondance 
179 
12.1 
The exterior algebra of a vector space 
180 
12.2 
The dual space 
183 
12.3 
Decomposable kvectors 
183 
12.4 
Creation and annihilation operators 
184 
12.5 
The Klein quadric 
187 
12.6 
The groups SL(V) and Ω(Λ_{2}(V)) 
190 
12.7 
Correlations 
191 
12.8 
Alternating forms and reflections 
195 
12.9 
Hermitian forms of Witt index 2 
196 
12.10 
Fourdimensional orthogonal groups 
199 
12.11 
Generalized quadrangles and duality 
201 
12.12 
The Suzuki groups 
202 
12.13 
Exercises 
207 




Bibliography 
213 

Index of Symbols 
221 

Index of names 
223 

Subject Index 
225 