  In this paper we achieve the quantization of a particle moving on the $SU(2)$
group manifold, that is, the three-dimensional sphere $S^{3}$, by using
group-theoretical methods. For this purpose, a fundamental role is played by
contact, non-point symmetries, i.e., symmetries that leave the
Poincar\'e-Cartan form semi-invariant at the classical level, although not
necessarily the Lagrangian. Special attention is paid to the role played by the
basic quantum commutators, which depart from the canonical, Heisenberg-Weyl
ones, as well as the relationship between the integration measure in the
Hilbert space of the system and the non-trivial topology of the configuration
space. Also, the quantization on momentum space is briefly outlined.
