Group theory

1
Group theory
Isomorphism Two groups isomorphic if they have
same type of multiplication table
2
Group theory
Isomorphism Two groups isomorphic if they have
same type of multiplication table
3
Group theory
Isomorphism Two groups isomorphic if they have
same type of multiplication table
4
Point Symmetry
Rotation axes 1 A A2 A3
Rotoinversion axes Aa i (Aa i)2
(Aa i)3 Aa i ( 1) Aa i Aa
Aa i Aa ( 1) Order of
improper cyclical rotation group always even
5
Point Symmetry
Rotation axes 1 A A2 A3
Rotoinversion axes Aa i (Aa i)2
(Aa i)3 Aa i ( 1) Aa i Aa
Aa i Aa ( 1)
Suppose n odd (3) Aa i Aa Aa i
Aa Aa i Aa ( 1) Aa i Aa
i Aa Aa i Aa ( 1)
6
Point Symmetry
Rotation axes 1 A A2 A3
Rotoinversion axes Aa i (Aa i)2
(Aa i)3 Aa i ( 1) Aa i Aa
Aa i Aa ( 1)
Suppose n odd (3) Aa i Aa Aa i
Aa Aa i Aa ( 1) Aa i Aa
i Aa Aa i Aa ( 1) 1 Aa
Aa g 3 h i i Aa i Aa i
g h 3
7
Point Symmetry

8
Point Symmetry
9
Point Symmetry
10
Point Symmetry
Subgroups G g 1
1 2 1 2 3 1
3 4 1 2 4 6
1 2 3 6
11
Point Symmetry
Subgroups G g 1
1 1 2 1 2 3
1 3 1 3 4 1 2
1 4 6 1 3 2 6

12
Point Symmetry
Group combinations C2 C2'
(222) 1 C2 1 1
C2 C2' C2' C2 C2' ( C2")
13
Point Symmetry
Group combinations C2 C2'
(32) 1 C3 C3 1
1 C3 C3 C2 C2 C2
C3 C2 C3 C2 C2' C2" transform 3
into itself
2
2
C2"
2
C2"
C2'
C2'
C2
14
Point Symmetry
Group combinations C2 C2'
(32) 1 C3 C3 1
1 C3 C3 C2 C2 C2
C3 C2 C3 C2 C2' C2" transform 3
into itself But C3 C2 C3 C2"
C3 C2' C3 C2 C3 C2" C3 C2'
2
2
C2"
2
C2"
C2'
C2'
C2
-1
-1
-1
15
Point Symmetry
Group combinations C2 C4
(422) 1 C4 C4
C4 1 1 C4
C4 C4 C2 C2 C2 C4
C2 C4 C2 C4
2
3
C2"'
2
3
2
3
C2"
C2"
C2'
C2"'
C2'
C2
16
Point Symmetry
Group combinations C3 C2 C2'
(3 222 23) 1 A2
B2 C2 D3 D3
D3 A2 D3 B2 D3 C2 D3 D3 D3
A2 D3 B2 D3 C2 1
A2 B2 C2
D3 D3
3E
C
1E
D
B
-1
-1
-1
2
-1
-1
A
2E
D
1E
Fill in rest of table
-1
A
17
Point Symmetry
Group combinations C3 C2 C2'
(3 222 23) 1 A2
B2 C2 D3 D3
D3 A2 D3 B2 D3 C2 D3 D3 D3
A2 D3 B2 D3 C2 1
A2 B2 C2
D3 1E3 2E3 3E3
D3 1E3 2E3
3E3
3E
C
1E
D
B
-1
-1
-1
2
-1
-1
A
2E
D
1E
-1
-1
-1
-1
A
18
Point Symmetry
Group combinations 2 23 432
1G
3E
C
5G
F
1E
3G
D
Do for homework
B
2G
A
4G
C
2E
1E
F
D
A