This document provides an overview of group theory concepts. A group is a collection of elements that is closed under a binary operation, contains an identity element, and has inverse elements. Groups can be represented by multiplication tables. Symmetry operations within a point group can be classified into conjugacy classes based on their similarity transforms. Matrix representations allow symmetry operations to be modeled as transformations on object coordinates.

1. Chapter 3 - Group Theory
A Group is a collection of elements which is:
i) closedunder some single-valuedassociative binary
operation
ii) contains a singleelement satisfyingthe identity law
iii) and has a reciprocalelement for each element in the
group
Collection: a specified# of elements (finiteor infinite)
Elements: the consitituentsof the group (i.e., symmetry
operations)
Binary Operation: the combinationof two elements of a
group to yield another element in the group. The
combinationmay be mathematical(addition, subtraction,
etc.) or qualitativeas in the successiveapplicationof two
symmetry operations on an object.
Single-valued: the combinationof two elements yields a
unique result
Closed: the combinationof any two group elements must
always yield another element belonging to the group.
Associative: the associativelaw of combinationmust hold
for the group.
(AB)C = A(BC)

2. Group Theory 2
In general, however, elements of a group do not have to
commute (but they can):
AB ≠ BA
Identity Law: there must be an element in the group which
when combined with any element in the group will leave
them unchanged. This element is calledthe identity or unit
element and it commutes with all elements of the group. It
is given the symbol E.
EA = A AE = A EE = E
Reciprocal Element: for each element A in a group there
must be an element calledthe reciprocal, A1, such that the
followingholds:
AA1 = A1A = E
In general, group multiplicationis not commutative,i.e.,
AB ≠ BA. However, it can be and a group in which
multiplicationis completely commutativeis called an
Abelian Group.

3. Group Theory 3
Group MultiplicationTable (matrix operations)
G3 E A B
E E A B
A A B E
B B E A
Each row and each column in a group multiplicationtable
lists each of the group elements ONCE and ONLY ONCE.
It therefore follows that no two columns or rows may be
identical!
Considera “real” C3 table using symmetry elements:
C3 E C3 C3
2
E E C3 C3
2
C3 C3 C3
2 E
C3
2
C3
2 E C3
(column) × (row)
Abelian group

4. Group Theory 4
Considerthe two different ways we can set up a 4 × 4 table:
G4
1 E A B C
E E A B C
A A E C B
B B C E A
C C B A E
Note that each element times itself generates E.
G4
2 E A B C
E E A B C
A A B C E
B B C E A
C C E A B
Note that this group table aboveis cyclic, that is, the group
is generated by one element:
A = A A3
= C
A2
= B A4
= E

5. Group Theory 5
Note that the G3 (C3) example abovewas also cyclic.
There is only one group combinationpossiblefor the G5
group, which turns out to be cyclic as well:
G5 E A B C D
E E A B C D
A A B C D E
B B C D E A
C C D E A B
D D E A B C
Note the diagonal lining up of the elements in cyclic groups
(symmetry of a matrix sort).

6. Group Theory 6
SubGroups
A subgroup is a self-containedgroup of elements residing
within a larger group.
(integer)
subgroup)of(order
group)mainof(order
k
g
h

G6 E A B C D F
E E A B C D F
A A E D F B C
B B F E D C A
C C D F E A B
D D C A B F E
F F B C A E D
G3 E D F
E E D F
D D F E
F F E D

7. Group Theory 7
Classes
Assume that A and X are elements of a group and we
perform the followingoperation:
X1
AX = B
Where B is anotherelement in the group. B is then called
the similaritytransformof A by X. If this relationship
holds, then A and B are said to be conjugate.
The followingis true for elements that are relatedby
similaritytransforms:
1) Every element is conjugate with itself
A = X1
AX
(X may be equal to the identity element E)
2) If A is conjugate with B, then B is conjugate with A
Thus, if we have:
X1
AX = B
Then there must exist another element Y such that:
Y1
BY = A
3) Finally, if A is conjugate to both B and C, then B and C
must also be conjugate to each other.
A group of elements that are conjugate to one another is
calleda Class of Elements.

8. Group Theory 8
To determine which elements group together to form a
class you have to work out all the similaritytransforms for
each element in the group. Thosesets of elements that
transforminto one another are then in the same class.
Considerthe C3v symmetry point group “matrix”:
C3v E C3 C3
2
v
1 v
2 v
3
E E C3 C3
2
v
1 v
2 v
3
C3 C3 C3
2 E v
2 v
3 v
1
C3
2 C3
2 E C3 v
3 v
1 v
2
v
1 v
1 v
2 v
3 E C3 C3
2
v
2 v
2 v
3 v
1 C3
2 E C3
v
3 v
3 v
1 v
2 C3 C3
2 E
Lets determine the classes of symmetry operationsfor this
point group. Lets start with the similaritytransforms for
the vertical mirror planes:
v
1v
1v
11 = v
1
v
2v
1v
21 = v
3
v
3v
1v
31 = v
2

9. Group Theory 9
Let’s see how this works graphically:
v
2v
1v
21 = v
3

10. Group Theory 10
v
3v
1v
31 = v
2

11. Group Theory 11
C3v
1C31 = v
3

12. Group Theory 12
If we continuethese similaritytransforms we find that the
varioussymmetry operationsfor C3v break down into the
followingclasses:
E
C3, C3
2
v
1, v
2 , v
3
If we examine the character tables in Cotton we find that
the symmetry operationsare listed and grouped together in
these very same classes:
Corollary: the orders of all the classes must be integral
factors of the order of a group.
Order of a point group = # of symmetry operations

13. Group Theory 13
Matrix Operations
Considerthe followingmatrix:
a11 a12 a13 a14 . . . a1n
a21 a22 a23 a24 . . . a2n
a31 a32 a33 a34 . . . a3n
. . .
an1 an2 an2 an2 . . . amn
Character: sum of diagonal elements
In order to multiply two matrices they must be
conformable,i.e., to multiply matrix A by matrix B, the
number of columns in A must equal the number of rows in
B.
(a11 × b11) + (a12 × b21) = c11
11 12 13 14
21 22 23 24
31 32
11 12 13 14
11 12
21 2
21 22
2
31 32
2 4
3 34
3
3
2
b b b b
b
c c c c
b b b
c c c c
c c c c
a a
a a
a a
   
         
       
3 × 2 2 × 4 = 3 × 4
Row
Column
Row Column

14. Group Theory 14
The symmetry operationscan all be represented
mathematicallyas 3 × 3 square matrices.
To carry out the symmetry operation, you multiply the
symmetry operationmatrix times the coordinatesyou want
to transform. The x, y, z coordinatesare written in vector
format as a 3 × 1 matrix:
x
y
z
 
 
 
  
For example, the inversionoperationtake the general
coordinatesx, y, z to x, y, z. In matrix terms we
would write:
1 0 0
0 1 0
0 0 1
x
y
z
x
y
z
     
           
         



x(new) = (1)(x) + (0)(y) + (0)(z)
y(new) = (0)(x) + (1)(y) + (0)(z)
z(new) = (0)(x) + (0)(y) + (1)(z)

15. Group Theory 15
Symmetry OperationMatrices:
1 0 0
0 1 0
0 0 1
x
y
z
x
y
z
     
          
          
1 0 0
0 1 0
0 0 1
x x
y y
z z
      
            
           
1 0 0
0 1 0
0 0 1
x
y
z
x
y
z
     
          
         
1 0 0
0 1 0
0 0 1
x
y
zz
x
y
     
           
         


1 0 0
0 1 0
0 0 1
x
y
z
x
y
z
     
          
         


E
i
(xy)
(xz)
(yz)

16. Group Theory 16
cos sin 0
sin cos 0
0 0
'
1
'
x
y
z z
x
y
 
 
     
           
          
cos sin 0
sin cos 0
0 0 1
'
'
x
y
x
y
zz
 
 
     
           
          
Cn × h = Sn
cos sin 0 1 0 0
sin cos 0 0 1 0
0 0 1 0 0 1
 
 
   
      
      
Cn h
Cn
Sn

17. Group Theory 17
Group Representations
The set of four matrices that describe all of the possible
symmetry operations in the C2v point group that can act on
a point with coordinatesx, y, z is calledthe total
representation of the C2v group.
1 0 0 1 0 0 1 0 0 1 0 0
0 1 0 0 1 0 0 1 0 0 1 0
0 0 1 0 0 1 0 0 1 0 0 1
        
               
              
E C2 xz yz
Note that each of these matrices is block diagonalized, i.e.,
the total matrix can be broken up into blocks of smaller
matrices that have no off-diagonal elements between blocks.
These block diagonalizedmatrices can be broken down, or
reduced into simplerone-dimensionalrepresentationsof
the 3-dimensionalmatrix.
If we considersymmetry operations on a point that only has
an x coordinate(e.g., x, 0, 0), then only the first row of our
total representationis required:
C2v E C2 xz yz
 1 1 1 1 x

18. Group Theory 18
We can do a similarbreakdownof the y and z coordinates
to setup a table:
C2v E C2 xz yz
 1 1 1 1 x
 1 1 1 1 y
 1 1 1 1 z
These three 1-dimensionalrepresentationsare as simple as
we can get and are called irreduciblerepresentations.
There is one additional irreduciblerepresentationin the
C2v point group. Considera rotation Rz :
The identity operationand the C2 rotation
operationsleavethe directionof the rotation
Rz unchanged. The mirror planes, however,
reverse the directionof the rotation
(clockwiseto counter-clockwise), so the irreducible
representationcan be written as:
C2v E C2 xz yz
 1 1 1 1 Rz
4 Classes of symmetry operations =

19. Group Theory 19
4 Irreducible representations!!
Now lets considera case where we have a 2-dimensional
irreduciblerepresentation. Considerthe matrices for C3v
1 0 0 cos120 sin120 0 1 0 0
0 1 0 sin120 cos120 0 0 1 0
0 0 1 0 0 1 0 0 1
     
          
          
E C3 v
In this case the matrices block diagonalizeto give two
reduced matrices. One that is 1-dimensional for the z
coordinate,and the other that is 2-dimensionalrelating the
x and y coordinates.
Multidimensional matrices are represented by their
characters (trace), which is the sum of the diagonal
elements.
Since cos(120º) = 0.50, we can write out the irreducible
representationsfor the 1- (z) and 2-dimensional
“degenerate” x and y representations:
C3v E 2C3 v
 1 1 1 z
 2 1 0 x,y

20. Group Theory 20
As with the C2v example, we have anotherirreducible
representation(3 symmetry classes = 3 irreducible
representations)based on the Rz rotationaxis. This
generates the full group representationtable:
C3v E 2C3 v
 1 1 1 z
 2 1 0 x,y
 1 1 1 Rz

21. Group Theory 21
Character Tables
Schoenflies symmetry symbol
Mulliken Symbol Notation
1) A or B: 1-dimensional representations
E : 2-dimensional representations
T : 3-dimensional representations
2) A = symmetric with respect to rotationby the Cn axis
B = anti-symmetric w/respect to rotationby Cn axis
Symmetric = + (positive) character
Anti-symmetric =  (negative) character
Characters of
the irreducible
representations
Mulliken
symbols
x, y, z
Rx, Ry, Rz
Squares &
binary products
of the
coordinates

22. Group Theory 22
3) Subscripts 1 and 2 associatedwith A and B symbols
indicatewhether a C2 axis  to the principleaxis
produces a symmetric (1) or anti-symmetric(2) result.
If C2 axes are absent, then it refers to the effect of
vertical mirror planes (e.g., C3v)
4) Primes and double primes indicaterepresentations
that are symmetric (  ) or anti-symmetric (  ) with
respect to a h mirror plane. They are NOT used
when one has an inversioncenter present (e.g., D2nh or
C2nh).
5) In groups with an inversion center, the subscript“g”
(“gerade” meaning even) represents a Mulliken
symbol that is symmetric with respect to inversion.

23. Group Theory 23
The symbol “u” (“ungerade” meaning uneven)
indicates that it is anti-symmetric.
6) The use of numerical subscripts on E and T symbols
followsome fairly complicatedrules that will not be
discussedhere. Considerthem to be somewhat
arbitrary.
Square and Binary Products
These are higher order “combinations” or products of the
primary x, y, and z axes.

24. Group Theory 24
The Great Orthogonality Theorem
  '' ' '
*( ) ( )m m mmn n ni ij
i
nj
j
R R
l l
h
      

25. Group Theory 25
i (R)mn The element in the mth row and nth column of
the matrix correspondingto the operationR in
the ith irreduciblerepresentationi.
i (R)mn
* complex conjugateused when imaginary or
complex #’s are present (otherwiseignored)
h the order of the group
li the dimension of the ith representation
(A = 1, B = 1, E = 2, T = 3)
 delta functions, = 1 when i = j, m = m’, or n =
n’; = 0 otherwise
The different irreduciblerepresentationsmay be thought of
as a series of orthonormal vectors in h-space, where h is the
order of the group.

26. Group Theory 26
Because of the presence of the delta functions, the equation
= 0 unless i = j, m = m’, or n = n’. Therefore, there is only
one case that will play a direct role in our chemical
applications:
  0' '( ) ( )
R
nm j m ni R R    
  0' '( ) ( )
R
nm j m ni R R    
  ( ) ( )
R
i inm nm
i
R R
l
h
  
if i ≠ j
if m ≠ m’
n ≠ n’

27. Group Theory 27
Five “Rules” about IrreducibleRepresentations:
1) The sum of the squares of the dimensionsof the
irreduciblerepresentationsof a group is equal to the
order, h, of a group.
2
il h
For example, considerthe D3h point group:
l(A1’)2 + l(A2’)2 + l(E’)2 + l(A1”)2 + l(A2”)2 + l(E”)2
(1)2 + (1)2 + (2)2 + (1)2 + (1)2 + (2)2 = 12
2) The sum of the squares of the characters in any
irreduciblerepresentationis also equal to the order of
the group h.
2
( )i
R
R hg    
For example, for the E’ representationin D3h:
(E)2 + 2(C3)2 + 3(C2)2 + (h)2 + 2(S3)2 + 3(h)2
h = 12 (order of group)
g = # of symmetry
operations R in a class
Dimensions:
A or B = 1
E = 2
T = 3

28. Group Theory 28
(2)2 + 2(-1)2 + 3(0)2 + (2)2 + 2(-1)2 + 3(0)2 = 12
3) The vectors whose components are the characters of
two different irreduciblerepresentationsare
orthogonal.
0( ) ( )i j
R
g R R  
For example, multiply out the A2’ and E’ representations
in D3h:
1(1)(2) + 2(1)(-1) + 3(-1)(0) + 1(1)(2) + 2(1)(-1) + 3(-1)(0)
2 + (-2) + 0 + 2 + (-2) + 0 = 0
4) In a given representationthe characters of all
matrices belonging to operationsin the same class are
identical.
5) The number of irreduciblerepresentationsin a group
is equal to the number of classesin the group.