Example 7 Show that the point group C 4 is isomorphic with a group containing 1,
i,-1,-i as members.
Solution: Given: point group C 4 , one mathematical group containing 1, i, −1, −i as
members. Let us write the group multiplication tables and check one-to-one correspondence between their elements.
Group multiplication tables of C 4 and the elements 1, i, −1, −i are:
C 4
E
C 4
C
2
4 ¼ ðC 2 Þ
C
3
4
E
E
C 4
C
2
4
C
3
4
C 4
C 4
C
2
4
C
3
4
E
C
2
4
C
2
4
C
3
4
E
C 4
C
3
4
C
3
4
E
C 4
C
2
4
1
i
−1
−i
1
1
i
−1
−i
i
i
−1
−i
1
−1
−1
−i
1
−i
−i
−i
1
I
−1
Comparing their elements shows that they follow the one-to-one citation, that is,
C 4
C
2
4
C
3
4
C
4
4 ¼ E
ð
Þ
i
i
2
i
3
i
4
i
−1
−i
1
⟹ They are isomorphic
Example 8 Show that points groups C 2h , D 2 and C 2v are isomorphic.
Solution: Given: Three points groups are: C 2h , D 2 and C 2v .
Let us write the group multiplication tables and check the one-to-one correspondence between their elements. Group multiplication tables are:
C 2h
E
C 2
r h
-i
E
E
C 2
r h
−i
C 2
C 2
E
- i
r h
r h
r h
-i
E
C 2
-i
-i
r h
C 2
E
6.3 Group (Point) Representation of Symmetry Operations
245