## Classical groups: Course given in spring term 1965 at Yale University |

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1-dimensional abelian acts transitively algebra analogous anti-automorphism argument assume char K f characteristic space characterized clearly commutator subgroup conjugate define denote determined dimension dimK division ring duality Euclidian extreme involutions follows form a minimal GLn(K hence hermitian form homomorphism hyperbolic basis hyperbolic plane implies induces intersection invariant involutions in 3p2 isomorphism isotropic line isotropic points isotropic vectors kernel leaves x fixed Lemma Let f Let G Let H linear space linear transformation lines containing mapping matrix maximal totally isotropic MCS-set minimal couple Moreover non-degenerate noncommutative nonisotropic hyperplane nonisotropic lines nonisotropic subspace nonisotropic vector normal subgroup obtain orthogonal group permutation group pointwise fixed preserves orthogonality Projective Geometry proof of 5.2 prove quasi-reflection quaternion quaternion field result s-linear semi-linear skew hermitian SLn(K suppose symmetric symplectic basis totally isotropic subspace trace valued Un/Tn unitary transvections Witt's theorem yields