Symbolic description of symmetry operations

So far we used both geometrical and verbal tools to describe symmetry elements (e.g. plane, axis, center and translation) and operations (e.g. reflection, rotation, inversion and shift). This is quite convenient when the sole purpose of this description is to understand the concepts of symmetry. However, it becomes difficult and time consuming when these tools are used to work with symmetry, for example to generate all possible symmetry operations, e.g. to complete a group. Therefore, two other methods are usually employed  

We begin with the symbolic description of symmetry operations, which is based on the fact that the action of any symmetry operation or any combination of symmetry operations can be described by the coordinates of the resulting object(s). In this section we assume that the initial object has 

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Figure 1.46. Symmetry elements and symbolic description of symmetry operations in the point group symmetry 2/m. Here, the origin of the coordinate system coincides with the center of inversion.

This description formalizes symmetry operations by using the coordinates of the resulting points and, therefore, it is broadly used to represent both symmetry operations and equivalent positions in the International Tables for Crystallography (see Table 1.18). The symbolic description of symmetry operations, however, is not formal enough to enable easy manipulations involving crystallographic symmetry operations. 

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