Interaction of symmetry elements

So far we have considered a total of 10 different crystallographic symmetry elements, some of which were combinations of two simple symmetry elements either acting simultaneously or consecutively. The majority of crystalline objects, e.g. crystals and molecules, have more than one symmetry element. 

Symmetry elements and operations interact with one another producing new symmetry elements and symmetry operations, respectively. When applied to symmetry, an interaction means consecutive (and not 

The question is what converts object X into object X2 The only logical answer is there should be an additional symmetry operation. No. 3, that converts object X into object X2. 

The mirror plane is, therefore, a derivative of the two-fold rotation axis and the center of inversion located on the axis. The derivative mirror plane is perpendicular to the axis and intersects the axis in a way that the center of inversion also belongs to the plane. If we start from the same pyramid A and apply the center of inversion first (this results in pyramid D) and the twofold axis second (i.e. A - B and D C), the resulting combination of four symmetrically equivalent objects and the derivative mirror plane remain the same. 

This example not only explains how the two symmetry elements interact, but it also serves as an illustration to a broader conclusion deduced above any two symmetry operations applied in sequence to the same object create a third symmetry operation, which applies to all symmetrically equivalent objects. Note, that if the second operation is the inverse of the first, then the resulting third operation is unity (the one-fold rotation axis, 1). For example, when a mirror plane, a center of inversion, or a two-fold rotation axis are applied twice, all result in a one-fold rotation axis. 

---

Figure 1.16. Schematic illustrating the interaction of symmetry elements. A two-fold rotation axis and a center of inversion located on the axis (left) result in a mirror plane perpendicular to the axis intersecting it at the center of inversion (right).

Note that here and below x designates a generic binary combination law, and not multiplication. For example, applied to symmetry groups the combination law (x) is the interaction of symmetry elements, in other words it is their sequential application, as has been described in section 1.6. For groups containing numerical elements, the combination law can be defined as addition or multiplication. Every group is always closed, even a group, which contains an infinite number of elements. 

---