## Classical Groups and Geometric Algebra''Classical groups'', named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups. Thus the story begins, as Weyl suggested, with ''Her All-embracing Majesty'', the general linear group $GL n(V)$ of all invertible linear transformations of a vector space $V$ over a field $F$. All further groups discussed are either subgroups of $GL n(V)$ or closely related quotient groups. Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops the required geometric notions, albeit from an algebraic point of view, as the end results should apply to vector spaces over more-or-less arbitrary fields, finite or infinite. The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups. This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles. It is intended for graduate students who have completed standard courses in linear algebra and abstract algebra. The author, L. C. Grove, is a well-known expert who has published extensively in the subject area. |

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### Contents

Permutation Actions | 1 |

The Basic Linear Groups | 5 |

Bilinear Forms | 13 |

Symplectic Groups | 21 |

Symmetric Forms and Quadratic Forms | 31 |

Orthogonal Geometry char F 2 | 39 |

Orthogonal Groups char F 2 I | 45 |

OV V Euclidean | 59 |

Orthogonal Groups char F 2 II | 75 |

Hermitian Forms and Unitary Spaces | 85 |

Unitary Groups | 93 |

Orthogonal Geometry char F 2 | 113 |

Clifford Algebras char F 2 | 119 |

Orthogonal Groups char F 2 | 127 |

Further Developments | 151 |

List of Notation | 165 |

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abelian anisotropic assume automorphism called chapter char F Chevalley groups Choose a basis Choose a hyperbolic classical groups Clifford algebra commutative compatible with Q conjugate Corollary define denoted derived group E O(V exists extends F is finite field F Fix(r fixed hyperplane follows form Q geometry GL(V half.turn hence homomorphism hyperbolic pair hyperbolic plane hyperplane Im(r induction isometry isomorphic isotropic vector Ker(r kernel linear transformation nondefective quadratic space nondegenerate nonsingular Note orthogonal basis orthogonal group Proposition 3.2 pu,y quadratic form quadratic space relative representing matrix ru,a set H simple group SL(V SO(V SO(W Sp(V space of dimension space over F spinor norm SU(V symmetric bilinear form symplectic basis Theorem 5.2 transitive on PC transvection ui,vi unique unitary space vector space Witt Extension Theorem Witt index write