General Relations Among Symmetry Elements and Operations

9 GENERAL RELATIONS AMONG SYMMETRY ELEMENTS AND OPERATIONS 

We present here some very general and useful rules about how different kinds of symmetry elements and operations are related. These deal with the way in which the existence of some two symmetry elements necessitates the existence of others, and with commutation relationships. Some of the statements are presented without proof the reader should profit by making the effort to verify them. 

The product of two proper rotations must be a proper rotation. Thus, although rotations can be created by combining reflections (see rule 2), the reverse is not possible. The special case C2(jt)C2(y) = C2(z) has already been examined (page 30). 

When there is a rotation axis, C , and a plane containing it, there must be n such planes separated by angles of 2nl2n. This follows from rule 2. 

A proper rotation axis of even order and a perpendicular reflection plane generate an inversion center, that is C%,a = aC = C2o = crC2 = i. Similarly C i = zC n = C2i - iC2 a. 

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