Group Theory: An Application of Discrete Groups

In order to analyze energy levels, bonding, and spectroscopy, it is of great importance to be able to determine the symmetry of a molecule. The symmetry of a molecule or atom can be analyzed strictly from the molecule itself, or from the environment of the atom. Symmetry properties are classified according to different groups determined by sets of symmetry operations. Different elements are considered when studying symmetry, such as geometrical properties, reflections, and rotations. Symmetries are determined through mathematical tools; these tools are known as Group Theory. Two main distinctions must be made about group theory in terms of the ability of the group to rotate only a finite number of times, or when the rotations depend upon angles which vary in a continuous matter. The first case is known as discrete theory, or finite groups; and the second one is known as continuous, or Lie groups. In this paper, symmetry will be examined from a discrete point of view.