ðAB)C ¼ AðBCÞ
ð6:11Þ
3. Every group contains one element called the identity element E, such that
AE ¼ EA ¼ A
ð6:12Þ
4. Every element in the group has its inverse also in the group, such that
AA
À1
¼ A
À1 A ¼ E
ð6:13Þ
Solved Examples
Example 1 Show that the symmetry elements E, C 3 ; C
2
3 constitute a group. What
is its order?
Solution: Given: Three symmetry elements are: E, C 3 ; C
2
3 order of the group = ?
Since the three symmetry elements are independent, hence the order of the group
is 3. Now let us check the following conditions.
(i) To check the closure property, let us take products, that is,
E C 3 ¼ C 3
E C
2
3 ¼ C
2
3
C 3 C
2
3 ¼ C
1
3 C
2
3 ¼ C
3
3 ¼ E
(ii) To cheek the associative property, let us consider the triple products, that is,
C 3 C 3 C
2
3
À
Á ¼ C 3 C 3
ð
ÞC
2
3
LHS ¼ C 3 C 3 C
2
3
À
Á
= C 3 C
3
3 = C 3 E ¼ C 3
RHS ¼ C 3 C 3
ð
ÞC
2
3 ¼ C
2
3
À Á
C
2
3 ¼ C 3
) LHS = RHS
(iii) The given symmetry elements contain one identify element, E which leaves
the other members unchanged, that is,
E C 3 ¼ C 3 E ¼ C 3
E C
2
3 ¼ C
2
3 E ¼ C
2
3
6.3 Group (Point) Representation of Symmetry Operations
239