  The role of discrete (or point-group) symmetries in alpha-cluster nuclei is
discussed in the framework of the algebraic cluster model which describes the
relative motion of the alpha-particles. Particular attention is paid to the
discrete symmetry of the geometric arrangement of the alpha-particles, and the
consequences for the structure of the corresponding rotational bands. The
method is applied to study cluster states in the nuclei 12C and 16O. The
observed level sequences can be understood in a simple way as a consequence of
the underlying discrete symmetry that characterizes the geometrical
configuration of the alpha-particles, i.e. an equilateral triangle with D(3h)
symmetry for 12C, and a tetrahedron with T(d) symmetry for 16O. The structure
of rotational bands provides a fingerprint of the underlying geometrical
configuration of alpha-particles.
