Computations in classical groups

Abstract

In this thesis we develop algorithms similar to the Gaussian eliminationalgorithm in symplectic and split orthogonal similitude groups As an applicationto this algorithm we compute the spinor norm for split orthogonalgroups Also we get similitude character for symplectic and split orthogonalsimilitude groups as a byproduct of our algorithms Consider a perfect field k with char k 6 2 which has a non trivial Galoisautomorphism of order 2 Further suppose that the fixed field k0 has theproperty that there are only finitely many field extensions of any finite degree In this thesis we prove that the number of z classes in the unitary groupdefined over k0 is finite Eventually we count the number of z classes in theunitary group over a finite field Fq and prove that this number is same asthat of the general linear group over Fq provided q gt n newline newline