The applications of group theory can be subdivided generally into two broad areas: where the underlying dynamic laws of interaction and therefore all the resulting symmetries are known exactly; the other where these are as yet unknown and only the kinetic symmetries(i.e. those of the underlying space-time continues) can serve as a certain guide.
In the first area, group theoretical techniques are used essentially to exploit the known symmetries, either to simplify the numeric calculations or to draw exact qualitative conclusions. All (extra nuclear) atomic and molecular phenomena are believed to belong to this category. In the second major area, application of group theory proceeds essentially in the opposite direction. It is used to discover as much as possible of the underlying symmetries and, through them, learn about the physical laws of interaction, this area, includes all the aspects of nuclear structure and elementary particle theory.
Group theory in atomic spectroscopy
Group theory plays two roles in atomic spectroscopy. Its most obvious function is to take an advantage of the symmetry possessed by the Hamiltonian for the electrons of an atom. If, for example, the atom is free the Hamiltonian must be invariant under rotations of the reference frame centered at the nucleus. The rotation group in three dimensions enters in a direct and physically significant way. It underlies the theory of angular momentum, and is centered to those applications of group theory and atomic spectroscopy.
The second role of group theory dates from the work of Racah. The fundamental idea is to consider the single-electron eigen functions as basis functions for transformations of a much more general character than those of rotation groups in three dimensions.
Group Theory in Solid State Physics
Group theory has become a most useful tool in modern physics for systemizing the description of idealized process dealing with theoretical concepts such as mass, energy, charge, momentum, and angular momentum; for classifying states in the quantum theory of matter; and further, for simplifying numerical applications of physical laws. The symmetry of a physical system is due to idealization, such as closed systems, isotropic spaces, ideal gases, incompressible fluids, and perfect solids.
It is remarkable fundamental property of nature that idealizations of this kind are really meaningful. That is, conclusions that can be drawn from assumptions of symmetry will become relevant. This may be explained that the configuration of highest possible symmetry usually is the most favorable physical state.