A general introduction to groups

Abstract

In this chapter, we introduce the fundamental concepts of group theory with the focus on those properties that are of particular importance for crystallography. Among other examples, the symmetry groups of an equilateral triangle and of the square are used throughout to illustrate the various concepts, whereas the actual application to crystallographic space groups will be found in later chapters. Starting from basic principles, we proceed to subgroups and the coset decomposition with respect to a subgroup. A particular type of subgroup is a normal subgroup. These are distinguished by the fact that the cosets with respect to such a subgroup can themselves be regarded as the elements of a group, called a factor group. These concepts have a very natural application to crystallographic space groups, since the translation subgroup is a normal subgroup and the corresponding factor group is precisely the point group of the space group. We then show how groups can be related by introducing homomorphisms, which are mappings between the groups that are compatible with the group operation. An important link between abstract groups and groups of symmetry operations is the notion of a group action. This formalizes the idea that group elements are applied to objects like points in space. In particular, objects that are mapped to each other by a group element are often regarded as equivalent and the subgroup of group elements that fix an object provides an important characterization of this object. Applied to crystallographic space groups acting on points in space, this gives rise to the concept of Wyckoff positions. We finally look at the notion of conjugacy and at normalizers, which provide important information on the intrinsic ambiguity in the symmetry description of crystal structures.