(ii) Important Properties of a Group
Cyclic Group
A group is said to be cyclic if all elements of the group can be generated by one
element, such as A, A
2 , A
3 , … A
n (=E). The element A is called the generator of the
group, where n refers to the total number of elements in the group and is called the
order of the group. Cyclic groups are abelian but the converse is not true. The point
groups C 2 (2), C 3 (3), C 4 (4) are C 6 (6) are example of cyclic groups.
Abelian Group
A group is said to be abelian if all its elements commute with one another. Further,
two elements A and B are said to commute with one another if AB = BA. In abelian
groups, each element is in a class by itself, since
XAX
À1
¼ AXX
À1
¼ AE ¼ A
Order of the Group
In general, it is the number of non-equivalent symmetry elements in the group. For
example, the point group C 2h (2/m) has four non-equivalent symmetry elements E
(1), C 2 (2),
`
r h (m) and S 2 ( 1). Hence, the order of this point group 4.
Example 4 Out of 32 point groups how many belong to cyclic group. Write the
member of each group in their increasing order.
Solution: Given: Thirty-two point groups; No. of cyclic groups and their
members = ?
Ten cyclic groups of order,
h ¼ 1 is : C 1
h ¼ 2 are: C 2 ; Ci and C s
h ¼ 3 is: C 3
h ¼ 4 are: C 4 and S 4
h ¼ 6 are: C 6 ; S 3 ¼ C 3h ð 6Þ and S 6 ¼ C 3i ð 3Þ
Members of C 1 : E
C 2 : E; C 2
C i : E; i
C s : E; r h
C 3 : E; C 3 ; C
2
3
C 4 : E; C 4 ; C
2
4 ¼ C 2
ð Þ; C
3
4
C 6 : E; C 6 ; C
2
6 ¼ C 3
ð Þ; C
3
6 ¼ C 2
ð Þ; C
4
6 ¼ C
2
3
À Á ; C
5
6
242
6 Unit Cell Symmeteries and Their Representations