Definitions

Group theory is a very broad field of study. We shall look only at a narrow part of the field. We are concerned here with the application of group theory to the analysis of physical and chemical systems. For our discussions a group consists of elements (or operators) that mathematically represent operations that leave a system in an equivalent state. For our purposes group multiplication is the sequential application of symmetry operations or the multiplication of square matrices representing two symmetry operations. The point group of interest in the analysis of atoms, molecules, and solids is the covering group, which consists of the elements (or operators) of rotation, reflection, and inversion under which the atom, molecule, or solid remains invariant. For crystalline solids the group (space group) is enlarged to include rotations, reflections, and inversion combined with translations under which the crystalline solid remains invariant.

A group is a collection of distinct elements that possess the following four characteristics.

1. Closure. The product of any two elements is an element of the group. If A and B ∈ G and AB = C, then C ∈ G (the symbol ∈ means “belongs to” or “is a member of the set that follows”).

2. Every group must contain the identity element, E, which commutes with all elements of G: EA = AE = A for all A ∈ G.

3. Elements of the group obey the associative law: A(BC) = (AB)C.

4. Each element has an inverse. If A ∈ G, then A−1 ∈ G, where AA−1 = A−1 A = E.