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 char k 6= 2, which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field k0 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 k0 is finite. Eventually, we count the number of z-classes in the unitary group over a finite field Fq, and prove that this number is same as that of the general linear group over Fq (provided q > n).