Computations in Classical Groups

Abstract

In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we get similitude character for symplectic and split orthogonal similitude groups, as a byproduct of our algorithms. Consider a perfect field k with odd characteristics, which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field k_0 has the property 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 group defined over k_0 is finite. Eventually, we count the number of z-classes in the unitary group over a finite field F_q and prove that this number is same as that of the general linear group over F_q when q is large enough.