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).
