  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.
