may be used to find the classes of symmetry elements/operations in a group. This
method consists of the following steps. They can be considered as rules.
1. If a symmetry element/operation commutes with all other symmetry (elements/
operations), then it is in a separate class. For example, the element E (1), C s, r h
or m h and C 2 (2) belong to separate classes.
2. A rotation operation (proper or improper) and its inverse belong to the same
class if there are n vertical mirrors or n perpendicular C 2 (2) axes.
3. Two rotations (proper or improper) about different axes belong to the same class
if there is a mirror operation that interchanges points on these two axes.
4. Two reflections through two different mirrors belong to the same class if there is
another operation which interchanges points on the two mirror planes.
Example 9 Show that the symmetry elements E, C 2 , r xz , r yz of the point group
C 2v (mm2) are members of the same class or not.
Solution: Given: Point group C 2v (mm2), symmetry elements are: E, C 2 , r xz , r yz
classes = ?
We have seen above that all these symmetry elements are inverses of their own.
Now, applying similarity transform on the symmetry element C 2 and referring the
group multiplication table of C 2v (mm2) we get
E C 2 E ¼ E C 2 ¼ C 2
C 2 C 2 C 2 ¼ C 2 E ¼ C 2
r xz C 2 r xz ¼ r xz r yz ¼ C 2
r yz C 2 r yz ¼ r yz r xz ¼ C 2
⟹ All similarity transforms generated the same symmetry element C 2 and hence
it forms a class of its own.
Similarly, applying similarity transforms on other symmetry operations, we
obtain
E r xz E ¼ E r xz ¼ r xz
C 2 r xz C 2 ¼ C 2 r xz ¼ r xz
r xz r xz r xz ¼ r xz E ¼ r xz
r yz r xz r yz ¼ r xz C 2 ¼ r xz
and
Er yz E ¼ Er yz ¼ r yz
C 2 r yz C 2 ¼ C 2 r xz ¼ r yz
r xz r yz r xz ¼ r xz C 2 ¼ r yz
r yz r yz r yz ¼ r yz E ¼ r yz
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6 Unit Cell Symmeteries and Their Representations