element, such that R-1 A R = B or A = R-1 B R
Elements in a group which are conjugate to each other form a conjugacy class.
Example
In C3V group, the elements are grouped into three conjugacy classes,
(R B ) ik bjaji R a k
1 1 i k j 1 1 1 n n n
1 (ajia ) kj ik b j k i 1 1 1
n n n
b jkj kk (B b ) k
j k 1 1 k 1
b c a
c a b
2. Conjugacy Class
In Group G, element B is said to be conjugated to element
A, if there exists another element R ( R is not the identity
n
n
n
Section 4.4 Classification of Point Group 0. Generation Elements 1. C1, Cs 2. Abelian Groups (Cn, Sn) 3. Cnv, Cnh, Cv 4. Dn, Dnh, Dnd, Dh 5. Td, Oh, and Ih
Example II and III are finite groups, Example I is an infinite group.
The Abelian Group (阿贝尔群)
The group multiplication is commutative. AB=BA
Example I and II are Abelian Groups, Example III is Non-Abelian
24 n 1 3 n 1

Subgroup Cn/2
When n=4k+2, there exists an inversion center
1 i ( S 42kk 2 )
S2 = {I, i (S21)} S4 = {I, S41 , C2(S42 ), S43}
Notation: Group Ci
Sn = {Sn1 , Sn2 , Sn3 ,…, Snn = I}
For even powers
S C h n (n ) C n
2 k 2 k 2 k
For odd powers
S n
21 k
nh ( C)
21 k
n C h
21 k
S n n. n ICn ,, C n , h. n h C. , ,, .C , C C . . n h
The Subgroup (子群)
A group (K) formed by a subset of the elements of Group G is called a subgroup of G. h(G)/h(K) = Integer
{I, C31, C32} is a subgroup of {I, C31, C32, a, b, c} {I, a} is a subgroup of {I, C31, C32, a, b, c} {I, b} is a subgroup of {I, C31, C32, a, b, c} {I, c} is a subgroup of {I, C31, C32, a, b, c}
Cs Group {I, }
Generator:
Cn Group {I, Cn1, Cn2 , …, Cnn-1}
Generator: Cn
Sn Group, Another Cyclic Group
Sn Group, the group generated by the
improper-rotation axis.
The Closure
The multiplication table of C3V
A
AB I C31 2 B C3
I I C31 C32
C31 C31 C32 I
C32 C32 I C31
a a c b
I C32 C31
b b a c
C31 I C32
c c b a
C32 C31 I
(Cn1)n = Cnn =I
The Cyclic Group is Abelian
Cni Cnj = (Cn1)i+j = (Cn1)j+i = Cnj Cni
The Inverse elements in the cyclic group
()-1 =
(Cni)-1 = Cnn-i
Cs and Cn Group, two Cyclic Groups
Section 4.3 The Group and its Representation
1. Definition 2. Conjugacy class (共轭类) 3. Character of group elements (表示) 4. Irreducible representation (不可约表示) 5. Character Table (特征标表) Section 4.4 Classification of Point Group
I
R-1 I R = R-1 R = I
C31, C32
a C31 a = C32
a , b, c
C31 a C32 = b , a b a = c
3. Character of Group Elements
Definition: The character of group element R is the trace of its representation matrix.
I + ( -I )=0
Example II
i 2 l
N complex numbers f (l ) e
, l = 0, 1, …, N-1 forms a GROUP if the operation is defined as “a multiplication” The closure: f(I) * f(J) = f(I+J) The identity element: f(0)=1, f(0) * f(I) = f(I) * 0 =f(I) The associate element: f(I)*f(J)*f(K)=(f(I)*f(J))*f(K) =f(I)*(f(J)*f(K))=f(I+J+K) The inverse element: the inverse element of f(I )is f(N-I) f(I) * f((N-I )=f(N)=f(0)
Section 4.3 The Group and its Representation
1. Definition 2. Conjugacy class (共轭类) 3. Character of group elements (表示) 4. Irreducible representation (不可约表示) 5. Character Table (特征标表)
Proof:
A) k ( B ab i k i
i 1k 1
n
n
Identical
B) k ( A ba i k i
i 1k 1
n
n
Property II:
Conjugated elements have equal value of character.
Proof:
1. Definition
A set of elements G{ A, B, C, …}, satisfy, (1) The closure: for any AG and B G, there exists AB G. (2) The identity element: I G, for any A G, IA=AI. Then I is the identity element of set G.
S6 = {I, S61 , C3(S62 ), i (S63), C32(S64 ), S65}
Case 2: Sn Group, n is an odd number Order h = 2n
2. Cyclic Group (循环群)
Definition: A group is a cyclic group if every element can be expressed as a power (乘幂) of a single element.
{I, } {I, Cn1, Cn2 , …, Cnn-1} ()1 = ()2 = I (Cn1)i = Cni
Example
C3 , A subgroup of C3V
AB I C31 C32
I I C31 C32
C31 C31 C32 I
C32 C32 I C31
a a c b
I C32 C31
b b a c
C31 I C32
c c b a
C32 C31 I
a b c
a b c
(3) The associative law: A(BC)=(AB)C.
(4) The inverse element: For every element A G, there
exists an element A-1 G, so that AA-1=I.