  We construct the spectrum generating algebra (SGA) for a free particle in the
three dimensional sphere $S^3$ for both, classical and quantum descriptions. In
the classical approach, the SGA supplies time-dependent constants of motion
that allow to solve algebraically the motion. In the quantum case, the SGA
include the ladder operators that give the eigenstates of the free Hamiltonian.
We study this quantum case from two equivalent points of view.
