In this talk, we are going to review the construction of some transformation groups  on closed symplectic manifolds namely:  the symplectic diffeomorphism group , the Hamiltonian diffeomorphism group and their $C^0$ counterparts.  Precisely we investigate the $C^0$ extension of the flux homomorphism initially defined on the symplectic diffeomorphism group  as a tool to understand the geometric and algebraic properties the Hamiltonian diffeomorphism group.