Concepts of relative velocity

The central concept of the theory of relativity is the relativity of velocity. The velocity of a material body is not an intrinsic property of the body; it depends on a free choice of reference system. Relative velocity is thus reference-dependent, it is not an absolute concept. We stress that even zero-velocity must be relative. Every reference system possesses its own zero-velocity relative only to that particular reference system. Does the theory of relativity formulated in terms of relative velocities, with many zero-velocities, imply the Lorentz isometry group? We discuss the many relative spaces of Galileo and Poincare, as quotient spaces.