Preface to the Second Edition Preface Chapter 1 Affine Geometry over Finite Fields 1.1 Vector Spaces and Matrices over Finite Fields 1.2 Affine Spaces and Affine Groups over Finite Fields 1.3 Quadrics in AG(n,Fq)for q Odd 1.4 Quadrics in AG(n,Fq)for q Even 1.5 Comments 1.6 Exercises Chapter 2 Projective Geometry over Finite Fields 2.1 Projective Spaces and Projective Groups over Finite Fields 2.2 Quadrics in PG(n,Fq) 2.3 Quadrics in PG(2,Fq) 2.4 Ovals in PG(2,Fq) 2.5 Comments 2.6 Exercises Chapter 3 Geometry of Symplectic Groups over Finite Fields 3.1 Symplectic Groups and Symplectic Spaces over Finite Fields 3.2 Anzahl Theorems in Symplectic Geometry over Finite Fields 3.3 Singular Symplectic Geometry over Finite Fields 3.4 Symplectic Polarities and Polar Spaces Arising from Then 3.5 Comments 3.6 Exercises Chapter 4 Geometry of Pseudo-symplectic Groups over Finite Fields of Characteristic 2 4.1 Pseudo-symplectic Groups over Finite Fields of Characteristic 2 4.2 Pseudo-symplectic Geometry over Finite Fields of Characteristic 2 4.3 Singular Pseudo-symplectic Geometry over Finite Fields of Characteristic 2 4.4 Pseudo-symplectic Geometry over Finite Fields of Characteristic 2 Again 4.5 Pseudo-symplectic Polarities 4.6 Comments 4.7 Exercises Chapter 5 Geometry of Unitary Groups over Finite Fields 5.1 Unitary Geometry over Finite Felds 5.2 Anzahl Theorems in Unitary Geometry over Finite Fields 5.3 Singular Unitary Geometry over Finite Fields 5.4 Unitary Polarities and Polar Spaces Arising from Them 5.5 Hermitian varieties in PG(N,Fq2)and AG(n,Fq2) 5.6 Comments 5.7 Exercises Chpater 6 Geometry of Orthogonal Groups over Finite Fields of Odd Characteristic 6.1 Orthogonal Geometry over Finite Felds of Odd Characteristic 6.2 Anzahl Theorems in Orthogonal Geometry over Finite Fields of Odd Characteristic 6.3 Singular Orthogonal Geometry over Finite Fields of Odd Characteristic 6.4 Orthogonal Polarities and Polar Spaces Arising from Them 6.5 Comments 6.7 Exercises Chapter 7 Geometry of Orthogonal Groups over Finite Fields of Characteristic 2 7.1 Orthogonal Geometry over Finite Fields of Characteristic 2 7.2 Anzahl Theorems in Orthogonal Geometry over Finite Fields of Characteristic 2 7.3 Singular Orthogonal Geometry over Finite Fields of Characteristic 2 7.4 Polar Spaces Arising from Orthogonal Geometry over Finite Fields of Characteristic 2 7.5 Comments 7.6 Exercises Appendix A Axiomatic Projective Geometry A.1 Incidence Structures A.2 Axioms of Projective Planes and the Principle of Duality A.3 Finite Projective Planes A.4 Collineations in Projective Planes A.5 Desargues Plause A.6 Projective Spaces Appendix B Polar Spaces and Generalized Quadrangles B.1 Polar Spaces B.2 Generalized Quadrangles Appendix C Critical Problems C.1 A Critical Problem in Finite Vector Spaces C.2 Critical Problems in Finite Unitary Spaces C.3 A Critical Problem in Finite Symplectic Spaces C.4 Comments Appendix D Moor-Penrose Generalized Inverses of Matrices over Finite Fields D.1 Definition and Properties D.2 Construction of Enumeration of M-P Invertible Matrices D.3 Generalization to Matrices over Fq2 D.4 Comments Appendix E Representations of Forms by Forms in a Finite Field E.1 Representations of Bilinear Forms E.2 Representations of Alternate Forms E.3 Representations of Hermitian Forms E.4 Representations of Quadratic Forms(Odd Characteristic Case) E.5 Representations of Quadratic Forms(Even Characteristic Case) E.6 Representations of Symmetric Bilinear Forms over Finite Fields of Even Characteristic E.7 Comments E.8 Exercises Bibliography Notation Index
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