This monograph is a comprehensive survey of the results obtained on the geometry of classical groups over finite fields mainly in the 1960s and early 1990s.For the convenience of the readers I start with the affine geometry and projective geometry over finite fields in Chapters 1 and 2, respectively.Among other things, the affine classification of quadrics is included in Chapter 1, and conics and ovals are studied in detail in Chapter 2. From Chapter 3 and onwards the geometries of symplectic, pseudo-symplectic,unitary, and orthogonal groups are studied in succession. The book ends with two appendices, on the axiomatic projective geometry, and on polar spaces and finite generalized quadrangles, respectively.

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 Conics in PG(n, Fq)

  2.4 Ovals in PG(n, 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 Them

  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 Fields

  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

Chapter 6 Geometry of Orthogonal Groups over Finite Fie.ids of Odd Characteristic

  6.1 Orthogonal Geometry over Finite Fields 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

  6.6 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 Planes

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