Definitions

All crystals and most molecules possess symmetry, which can be exploited to simplify the discussion of their physical properties. Changes from one configuration to an indistinguishable configuration are brought about by sets of symmetry operators, which form particular mathematical structures called groups. We thus commence our study of group theory with some definitions and properties of groups of abstract elements. All such definitions and properties then automatically apply to all sets that possess the properties of a group, including symmetry groups.

Binary composition in a set of abstract elements {gi}, whatever its nature, is always written as a multiplication and is usually referred to as “multiplication” whatever it actually may be. For example, if gi and gj are operators then the product gi gj means “carry out the operation implied by gj and then that implied by gi.” If gi and gj are both n-dimensional square matrices then gi gj is the matrix product of the two matrices gi and gj evaluated using the usual row × column law of matrix multiplication. (The properties of matrices that are made use of in this book are reviewed in Appendix A1.) Binary composition is unique but is not necessarily commutative: gi gj may or may not be equal to gj gi.