Symplectic GroupsSequel to the author's Lectures on Linear Groups, this book describes isomorphism theory of symplectic groups over integral domains. It presents the description of the isomorphisms of the symplectic groups and their congruence subgroups over integral domains. |
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abstract vector space apply assume base x1 big dilation bijection char F characteristic Chevalley groups conjugate contains defined denote EPGSp equation exceptional automorphism F is perfect field F fixed space follows fractional ideal geometric transformation GL(M GL(V group isomorphism group of symplectic GSpa hence hyperplane implies integral domain invertible involution isometry Let tē let us show linear groups Linear Lectures linear transformation Math matrix mean any subgroup nonorthogonal nontrivial projective transvection nontrivial transvection nonzero integral ideal normal subgroup o-module orthogonal perfect field PGL(V PGSp preserves projective transvections projective hyperbolic transformations projective symplectic collinear PROOF PTSp,(V PTSpa q(ox regular alternating space regular plane residual line residual space reso Spa(V Suppose symmetric matrix symplectic base symplectic collinear transformation symplectic group totally degenerate plane totally degenerate subspace transformation g transvections in Sp,(V transvections in Spa TSpa underlying dimension Witt's Theorem