Symmetry, Point Groups and Generators

A point group consists of operations that leave a single point invariant. These operations are rotations, inversion and reflections. The various points groups are formed by combining the operators in various ways. The derivation of all the point groups in a systematic way was done by Seitz1 A Here we shall only list them in a systematic way and discuss the set of symmetry operations that may be used to generate them. 

A word about notation. For the points groups we shall generally use the Schonflies notation. In this notation basically the same symbols are used for the point groups as for the symmetry operations. Whenever there is any doubt as to the meaning we shall use a caret (example Cn) to indicate a symmetry operation. 

The simplest point groups are generated by just one element. When this element is a rotation around some axis through an angle of 2 Jt/n we have a rotation operation of order n, symbolized Cn. The point group, Cn, consists of this operation and its various powers. 

The single generating operation can also be a rotation reflection operation, S2n, consisting of a rotation through 2nl2n followed by a reflection in a plane perpendicular to the rotation axis. The group is S2n. The operation S2 is identical to the inversion operation, I, and for n odd S2n can also be generated by the two operations Cn and I. 

---