TABLE OF CONTENTS Since the purpose of this section is to introduce group theory terminology and notations, no effort will be made to prove many of the statements and theorems mentioned in it. At most we shall illustrate our statements with examples and refer the reader to books on group theory for rigorous proofs.
The first step in studying the symmetry properties of any crystal is to determine its symmetry operations. For example: a square is unchanged under reflection about its two diagonals, or under rotation by 90 about an axis perpendicular to the square and passing through its center. One can generate other symmetry operations for a square which are combinations of these operations. One may say that it is possible to find an infinite number of symmetry operations for this square. However, many of these symmetry operations can be shown to consist of sequences of a few basic symmetry operations. The mathematical tool for systematically analyzing the symmetry operations of any object is group theory.
A group G is defined as a set of elements { a, b, c, } for which an operation ab (which we will refer to as multiplication) between any two elements a and b of the group is defined. This operation must have these four properties:
Closure: The result of the operation ab on any two elements a and b in G must...
