  In this article, we prove commutativity principal for linear, symplectic and
transvection groups. This principle is a consequence of Quillen-Suslin local
global principle and using a non-symmetric application of it as done by A. Bak.
The existence of a Local-Global Principle enables us to prove similar results
in various groups. We restrict ourselves to the classical symplectic,
orthogonal groups (and their relative versions); and to the automorphism groups
of a projective module (with a unimodular element), a symplectic module (with
ahyperbolic summand), and an orthogonal module (with a hyperbolic symmand).
  We could show that the symplectic quotients were abelian, but we could only
establish that the orthogonal quotients are solvable of length atmost two. We
do believe that the orthogonal quotient groups are also abelian; and prove this
when the base ring is a regular local ring containing a field.
