 In general, if G is a symmetry group of a theory describing a physical system (that is, the dynamical equations of the theory are invariant under the transformations of G), this means that the states of the system transform into each other according to some “representation” of the group G. In other words, the group transformations are mathematically represented in the state space by operations relating the states to each other. In quantum mechanics, these operations are implemented through the operators that act on the state space and correspond to the physical observables, and any state of a physical system can be described as a superposition of states of elementary systems, that is, of systems the states of which transform according to the “irreducible” representations of the symmetry group. Quantum mechanics thus offers a particularly favourable framework for the application of symmetry principles. The observables representing the action of the symmetries of the theory in the state space, and therefore commuting with the Hamiltonian of the system, play the role of the conserved quantities; furthermore, the eigenvalue spectra of the invariants of the symmetry group provide the labels for classifying the irreducible representations of the group: on this fact is grounded the possibility of associating the values of the invariant properties characterizing physical systems with the labels of the irreducible representations of symmetry groups, i.e. of classifying elementary physical systems by studying the irreducible representations of the symmetry groups.