Symmetry theory in a two-level quantum system

We develop the theory of symmetry for a two-level quantum system in oder to illustrate the main ideas of the general theory of symmetry in quantum theory. It is based on the diffeomorphism of the two-dimensional sphere S² onto the space of states P¹ and the isomorphism between the groups P(2) and SO₃(). In particular, rotational invariance leads to the appearance of the spin1/2 in a natural way.