5 Group Theory

7/31/2019 5 Group Theory

1/78

Group Theory

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I

y Inverse of product

yFurther Properties

yFurther Properties 2

yExample 2yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

Equiv Relation, Class

Congruence & Cosets

Paul Lim Group Theory 1 / 78


2/78

Introduction

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties



Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider the sets of operations, such as rotations, reflections and

inversions which transform physical objects (e.g. molecules) into

physically indistinguishable copies of themselves so that only the

labeling of identical components of the system (the atoms)

changes in the process.


3/78

Introduction 2

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties



Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


FIG. 1: (a) The hydrogen

molecule, and (b) the ammo-nia molecule.

The hydrogen molecule consists of

two atoms H of hydrogen, and is

carried into itself by any of the fol-

lowing operations:

(i) any rotation about its long

axis

(ii) rotation through about

an axis perpendicular to

the long axis and passing

through the point M that lies

midway between the atoms

(iii) inversion through the pointM

(iv) reflection in the plane that

passes through M and has

its normal parallel to the longaxis

These operations collectively form

a set of symmetry operations for

the hydrogen molecule.


4/78

Definition of a Group

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties



Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


A group G is a set of elements fX ; Y ; : : : g, together with a rule forcombining them that associates with each ordered pair X, Y a

product or combination law X Y for which the conditions must

be satisfied.

(i) For every pair of elements X, Y that belongs to G, the

product X Y also belongs to G (This is known as the

closure property of the group).

(ii) For all triples X, Y, Z the associative law holds i.e.

X .Y Z/ D .X Y / Z (1)


5/78

Definition of a Group 2

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties



Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


(iii) There exists a unique element I, belonging to G, with the

property that

I X D X D X I (2)

for all X belonging to G. This element I is known as theidentity element of the group.

(iv) For every element X of G, there exists an element X1, also

belonging to G such that

X1 X D I D X X1 (3)

X1

is called the inverse of X.

An alternative notation in common use is to write the elements of a

group G as the set fG1; G2; : : : g or fGi g.


6/78

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Finite group: the number of elements is finite (order of group g).

X Y denotes operation Y followed by X.

If any two particular elements of a group satisfy

Y X D X Y (4)

they are said to commute under the operation ; if all pairs ofelements in a group satisfy (4), then the group is said to be Abelian.

Eg: The set of all integers forms an Abelian group under addition.


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Example 1

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


ExampleUsing only the first equalities in (2) and (3), deduce the second

ones.

SolutionConsider the expression X1 .X X1/;

X1 .X X1/.ii/D .X1 X / X1

.iv/D I X1

.iii/D X1 (5)

But X1 belongs to G, and so from (iv) there is an element U in G

such that

U X1 D I .v/

Form the product of U with two extremes of (5) to give

U .X1 .X X1// D U X1 .v/D I: (6)


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Example 1 contd

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Transforming the LHS of this equation gives

U .X1 .X X1//.ii/D .U X1/ .X X1/

.v/

D I .X X1

/.iii/D X X1 (7)

Comparing (6) and (7) shows that

X X1 D I .iv/0

i.e. the second equality in group definition (iv). Similarly

X I.iv/D X .X1 X /

.ii/D .X X1/ X

.iv/0D I X

.iii/D X: .iii0/ i.e. the second equality in group definition (iii)


9/78

Uniqueness of Identity Element

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Suppose that I0, belonging to G, also has the property

I0 X D X D X I0 for all X belonging to G

Take X as I, thenI0 I D I (8)

Further from .iii0/,

X D X I for all X belonging to G

and setting X D I0 gives

I0 D I0 I (9)

It then follows from (8), (9) that I D I0, showing that in any

particular group the identity element is unique.


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Inverse of product

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider the product:

.U V Y Z/

.Z1 Y1 V1 U1/

D .U V Y / .Z Z1/

.Y1 V1 U1/

D .U V Y /

.Y1 V1 U1/

:::

D I

where use has been made of the associativity and of the equations

Z Z1 D I and I X D X.

) .U V Y Z/1 D .Z1 Y1 V1 U1/ (10)


11/78

Further Properties

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Further elementary results are

(i) Division Axiom: Given any pair of elements X, Y belonging

to G, there exist unique elements U, V also belonging to G,

such thatX U D Y and V X D Y

Clearly U D X1 Y and V D Y X1.

(ii) Cancellation Law: If

X Y D X Z

for some X belonging to G, then Y D Z. Similarly,

Y X D Z X

implies the same conclusion.


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Further Properties 2

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


(iii) Permutation Law: Forming the product of each element of G

with a fixed element X of G simply permutes the elements of

G; this is often written symbolically as G X D G. If this

were not so, and X Y and X Z were not different eventhough Y and Z were, application of the cancellation law

would lead to a contradiction.

Order of element: In any finite group of order g, any element X

when combined with itself to form successively X2 D X X,

X3 D X X2, . . . will, after at most g 1 such combinations,

produce the group identity I. If the number of combinationsneeded is m 1 i.e. Xm D I, then m is called the order of the

element X in G.


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Example 2

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


ExampleDetermine the order of the group of (two-dimensional) rotations

and reflections that take a plane equilateral triangle into itself, and

the order of each of the elements. The group is usually known as

3m (to physicists and crystallographers) or C3v (to chemists).

FIG. 2: Reflections in the three perpendicular bisectors of the sides of an

equilateral triangle take the triangle into itself.


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Example 2 contd

Group Theory

y Introduction

y Introduction 2

yDefinition

yDefinition 2

yExample 1

yExample 1 contd

yUniqueness of I


yFurther Properties


yExample 2

yExample 2 contd

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Solution

Two clockwise rotations, by 2=3 and 4=3, about an axis

perpendicular to the plane of the triangle

Reflections in the perpendicular bisectors of the three sides

Identity operation

In total, there are six distinct operations and g D 6 for this

group

Each rotation element of the group has order 3, and each reflection

element has order 2.


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Cyclic Group

Group Theory

Cyclic Group

yCyclic Group

yFurther Examples

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets



16/78

Cyclic Group

Group Theory

Cyclic Group

yCyclic Group

yFurther Examples

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


A so-called cyclic group is one for which all members of the groupcan be generated from just one element X (say). Thus a cyclic

group of order g can be written as

G D fI ; X ; X 2; X3; : : : ; X g1g

Cyclic groups are always Abelian. Also, each element, apart from

the identity, has order g, the order of the group itself, provided g is

a prime number.


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Further Examples of Groups

Group Theory

Cyclic Group

yCyclic Group

yFurther Examples

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Integers form a group under ordinary addition, but they do not

do so under ordinary multiplication. 1 is the identity, but the

inverse of any integer n, namely, 1=n, does not belong to the

set of integers.

Groups: sets of all real numbers, or of all complex numbers,

under addition, and of the same two sets excluding 0 under

multiplication. All of these groups are Abelian.

Complex numbers with unit modulus i.e. of the form ei

where 0 < 2 , form a group under multiplication:

ei1 ei2 D ei.1C

2/ .closure/ei 0 D 1 .identity/

ei.2/ ei D ei 2 ei 0 D 1 .inverse/


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Finite Groups

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table

yMultiplicatn Table 2


yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets



19/78

Finite groups: Example

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider the set S defined as

S D f1;3;5;7g under multiplication (mod 8)

Product (mod 8) of any two elements: multiply them together, andthen divide by 8, the remainder is the product of the two elements.

Since Y Z D Z Y, the full set of different products is

1 1 D 1; 1 3 D 3; 1 5 D 5; 1 7 D 7;

3 3 D 1; 3 5 D 7; 3 7 D 5;

5 5 D 1; 5 7 D 3;

7 7 D 1

Abelian group of order 4.


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Finite groups 2

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


It is convenient to present the results of combining any twoelements of a group in the form of multiplication tables.

1 3 5 7

1 1 3 5 7

3 3 1 7 5

5 5 7 1 3

7 7 5 3 1

Table 1: The table of products for the elements of the group

S D f1;3;5;7g under multiplication (mod 8).


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Multiplication Tables

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Some simple general properties of group multiplication tables:

1. Each element appears once and only once in each row or

column of the tablePermutation law: G X D G.

2. The inverse of any element Y can be found by looking along

the row in which Y appears in the left-hand column, and

noting the element Z at the head of the column in which the

identity appears as the table entry.When the identity appears on the leading diagonal, the

corresponding header element is of order 2 (unless it is the

identity).

3. For any Abelian group the multiplication table is symmetric

about the leading diagonal.


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Multiplication Tables 2

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider two sets of integers under multiplication (mod N):

S0 D f1;5;7;11g under multiplication (mod 24)

andS00 D f1;2;3;4g under multiplication (mod 5)

.a/

1 5 7 11

1 1 5 7 11

5 5 1 11 7

7 7 11 1 5

11 11 7 5 1

.b/

1 2 3 4

1 1 2 3 4

2 2 4 1 3

3 3 1 4 2

4 4 3 2 1TABLE. 2: On the left, the multiplication table for the group

S 0 D f1;5;7;11g under multiplication (mod 24). On the right,

multiplication is (mod 5).


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Multiplication Tables 3

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


I A B C

I I A B C

A A I C BB B C I A

C C B A I

TABLE. 3: The common structure exemplified by previous two tables.These two groups S D f1;3;5;7g and S 0 D f1;5;7;11g have

equivalent group multiplication tablesthey are said to be

isomorphic.


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Example 3

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


ExampleDetermine the behaviour of the set of four elements f1;i; 1; i g

under the ordinary multiplication of complex numbers. Show that

they form a group and determine whether the group is isomorphic

to either of the groups S (itself isomorphic to S 0 and S 00 defined

above.

Solution

1 i 1 i

1 1 i 1 i

i i 1 i 1

1 1 i 1 ii i 1 i 1

Table. 4: The group table for the set f1;i; 1; i g under ordinary

multiplication of complex numbers.

-

)


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Example 3 contd

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Form a group under associative operation of complex numbers.

Group f1;i; 1; i g is NOT isomorphic to S or S 0 (Table 3): Since

identity element 1 only appears on the leading diagonal twice

whereas I appears on the leading diagonal four times in Table 3,Table 4 cannot be brought into the form of Table 3.

1 i 1 i

1 1 i 1 ii i 1 i 1

1 1 i 1 i

i i 1 i 1

1 2 4 3

1 1 2 4 3

2 2 4 3 1

4 4 3 1 2

3 3 1 2 4

Table. 5: A comparison between Tables 4 and 2 (b), the latter with its

column reordered.


26/78

Example 3 contd

Group Theory

Cyclic Group

Finite Groups

yFinite Groups

yFinite Groups 2

yMultiplicatn Table



yExample 3

yExample 3 contd

yExample 3 contd

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


However, if the rows and columns of Table 2 (b) are rearranged,then the two tables can be compared (Table 5).

I A B C

I I A B C

A A B C I

B B C I A

C C I A B

TABLE. 6: The common structure exemplified by Tables 4 and 2(b), the

latter with its columns reordered.

They have the same structure as Table 6. Thus, the group

f1;i; 1; i g under ordinary multiplication of complex numbers is

isomorphic to the group f1;2;3;4g under multiplication (mod 5).


27/78

Non-Abelian Groups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


To be non-Abelian, a group needs at least 6 elements.


28/78

Non-Abelian groups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider as elements of a group the 2D operations whichtransform an equilateral triangle into itself. There are six such

operations; the null operation, two rotations (by 2=3 and 4=3

about an axis perpendicular to the plane of the triangle) and three

reflections in the perpendicular bisectors of the three sides.

Denote these operations by the following symbols:

1. I is the null operation.

2. R is a (clockwise) rotation by 2=3, and R0 that by 4=3.

3. K ;L;M are reflections in the three lines indicated in

Figure 2.


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Non-Abelian groups 2

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Thus,

R R D R0; R0 R0 D R; R R0 D I D R0 R

K K D L L D M M D I

(11)

Others, such as K M, can be found by drawing a sequence of

diagrams such as those following.

showing that K M D R0.


30/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Similarly,

shows that M K D R, and

shows that R L D K.


31/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


The multiplication table is shown below:

I R R0 K L M

I I R R

0

K L MR R R0 I M K L

R0 R0 I R L M K

K K L M I R R0

L L M K R0

I RM M K L R R0 I

Table. 7: The group table for the two-dimensional symmetry operations

on an equilateral triangle.


32/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


A number of things may be noticed from this table.

(i) It is not symmetric about the leading diagonal, indicating that

some pairs of elements in the group do not commute.

(ii) There is some symmetry within the 3 3 blocks that form the

four quarters of the table. This occurs because similar

operations are put close to each other when choosing the

order of table headingsthe two rotations, followed by thethree reflections.


33/78

Example 4

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider the set M of six orthogonal 2 2 matrices given by

ID

0@ 1 0

0 1

1A A D

0@ 12

p3

2

p3

2

1

2

1A B D

0@ 12

p3

2p3

2

1

2

1A

CD

0@ 1 0

0 1

1A D D

0@ 12

p3

2

p3

2

1

2

1A E D

0@ 12

p3

2p3

2

1

2

1A (12)

the combination law being that of ordinary multiplication. Note that

the matrices are group elements.


34/78

Example 4 contd

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Identity element: matrix I; I ; C ; D and E being their own inverses;inverse of A: B .

I A B C D E

I I A B C D EA A B I E C D

B B I A D E C

C C D E I A B

D D E C B I AE E C D A B I

Table. 8: The group table, under matrix multiplication, for the set M of six

orthogonal 2 2 matrices given by (12)

If fR; R0; K ; L ; M g of Table 7 are replaced by fA ; B ; C ; D ; Eg

respectively, the two tables are identical. The two groups are thus

isomorphic.

E l


35/78

Example 5

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Consider a set of functions of an undetermined variable x. Thefunctions are as follows:

f1.x/ D x; f2.x/ D 1=.1 x/; f3.x/ D .x 1/=x

f4.x/ D 1=x; f5.x/ D 1 x; f6.x/ D x=.x 1/

and the law of combination is

fi .x/ fj.x/ D fi .fj.x//

To calculate the product f6 f3: The product will be the function of

x obtained by evaluating y=.y 1/, when y is set equal to

.x 1/=x. Thus,

f6.f3/ D.x 1/x

.x 1/=x 1D 1 x D f5.x/

) f6 f3 D f5.

/

E l 5 d


36/78

Example 5 contd

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

yNon-Abelian

yNon-Abelian 2

yNon-Abelian 3

yNon-Abelian 4

yNon-Abelian 5

yExample 4

yExample 4 contd

yExample 5

yExample 5 contd

Permutation Groups

Mapping

Subgroups


Congruence & Cosets


Similarly,

f6 f6 Dx=.x 1/

x.x 1/ 1D x D f1 (13)

Note that if the symbols f1; f2; f3; f4; f5; f5; f6 are replaced by

I ; A ; B ; C ; D ; E respectively, the group table becomes identical to

Table 8.


37/78

Permutation Groups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation GroupsyPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


P t ti G


38/78

Permutation Groups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation GroupsyPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


The operation of rearranging n distinct objects amongstthemselves is called a permutation of degree n.

Example: The symmetry operations on an equilateral triangle can

be considered as the six possible rearrangements of the markedcorners of the triangle amongst three fixed points in space.

Example: The symmetry operations on a cube can be viewed as a

rearrangement of its corners amongst eight points in space or as arearrangement of its body diagonals in space.

NOTE: It is the permutations and not the objects (represented by

letters a ; b ; c ; : : : ), that form the elements of permutation groups.

P t ti N t ti


39/78

Permutation Notation

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


The complete group of all permutations of degree n is denoted bySn. The number of possible permutations of degree n is n (order

of Sn).

Suppose the ordered set of six distinct objectsfa b c d e fg

is

rearranged by some process into fb e f a d cg:

fa b c d e fg D fb e f a d cg;

where is a permutation of degree 6. The permutation can be

denoted by [2 5 6 1 4 3], since the first object a is replaced by the

second b, the second b replaced by the fifth e, the third by the sixth

f, etc.

The equation can then be written as

fa b c d e fg D 2 5 6 1 4 3fa b c d e fg

D fb e f a d cg:

P t ti N t ti 2


40/78

Permutation Notation 2

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


If is a second permutation, also of degree 6, then the product is

fa b c d e fg D .fa b c d e fg/:

Suppose that is the permutation [4 5 3 6 2 1]. then

fa b c d e fg

D 4 5 3 6 2 12 5 6 1 4 3fa b c d e fg

D 4 5 3 6 2 1fb e f a d cgD fa d f c e bg

D 1 4 6 3 5 2fa b c d e fg

Written in terms of the permutation notation this result is

4 5 3 6 2 12 5 6 1 4 3 D 1 4 6 3 5 2:

C l N t ti


41/78

Cycle Notation

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


For the permutation :the 1st object, a, has been replaced by the 2nd, b;

the 2nd object, b, has been replaced by the 5th, e;

the 5th object, e, has been replaced by the 4th, d;

the 4th object, d, has been replaced by the 1st, a;

This brings us back to the beginning of a closed cycle, which is

conveniently represented by the notation (1 2 5 4), in which the

successive replacement positions are enclosed, in sequence, inparentheses. Thus (1 2 5 4) means 2nd ! 1st, 5th ! 2nd, 4th !

5th, 1st ! 4th.

Cycle Notation 2


42/78

Cycle Notation 2

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


The remaining two objects, c and f, are interchanged by , ormore formally, are rearranged according to a cycle of length 2, or

transposition, represented by (3 6). Thus the complete

representation of is

D .1254/.36/

The positions of objects that are unaltered by a permutation are

either placed by themselves in a pair of parentheses or omittedaltogether.

Thus the identity permutation of degree is

I D .1/.2/.3/.4/.5/.6/;

though in practice it is often shortened to (1).

Cycle Notation 3


43/78

Cycle Notation 3

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


The order of any permutation of degree n within the group Sn isgiven by the lowest common multiple (LCM) of the lengths of the

cycles. Thus I has order 1, and the permutation discussed

above has order 4 (the LCM of 4 and 2).

Expressed in cycle notation our second permutation is (3)(1 4

6)(2 5), and the product is calculated as

.3/.1 4 6/.2 5/ .1 2 5 4/.3 6/fa b c d e fgD .3/.1 4 6/.2 5/fb e f a d cg

D fa d f c e bg

D .1/.5/.2 4 3 6/fa b c d e fg

Cycle Notation 4


44/78

Cycle Notation 4

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


Thus,

.3/.1 4 6/.2 5/ .1 2 5 4/.3 6/ D .1/.5/.2 4 3 6/

has order 6 (the LCM of 1, 3 and 2) and has order 4.

Consider the following group:

I D .1/.2/.3/ A D .1 2 3/ B D .1 3 2/

C D .1/.2 3/ D D .3/.1 2/ E D .2/.1 3/

A and B have order 3, whilst C, D and E have order 2. Their

combination products are exactly those corresponding to Table 8,with I, C, D, and E being their own inverses.

Cycle Notation 5


45/78

Cycle Notation 5

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

yPermutatn Groups

yPermutatn Notatn

yPermutatn Notatn 2

yCycle Notation

yCycle Notation 2

yCycle Notation 3

yCycle Notation 4

yCycle Notation 5

Mapping

Subgroups


Congruence & Cosets


For example

D Cfa b cg D .3/.1 2/ .1/.2 3/fa b cg

D .3/.1 2/fa c bg

D fc a bg

D .3 2 1/fa b cg

D .1 3 2/fa b cg

D Bfa b cg

The six permutations belonging to S3 form another non-Abelian

group isomorphic to rotation-reflection symmetry group of anequilateral triangle.


46/78

Mapping between Groups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

yMapping

yMapping 2

yMapping 3

yMapping 4

yMapping 5

Subgroups


Congruence & Cosets


Mappings between Groups


47/78

Mappings between Groups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

yMapping

yMapping 2

yMapping 3

yMapping 4

yMapping 5

Subgroups


Congruence & Cosets


When there is no ambiguity we will write the product of twoelements, X Y, simply as X Y.

If G and G0 are two groups, we can study the effect of a mapping:

W G ! G0

of G onto G0. If X is an element of G we denote its image in G0

under the mapping by X0

D .X/.

Mappings between Groups 2


48/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

yMapping

yMapping 2

yMapping 3

yMapping 4

yMapping 5

Subgroups


Congruence & Cosets


Let G and G0 be two groups and a mapping of G ! G0. If for

any pair of elements X and Y in G

. X Y /0

D X0

Y0

then is called a homomorphism, and G0 is said to be

homomorphic image of G.



49/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

yMapping

yMapping 2

yMapping 3

yMapping 4

yMapping 5

Subgroups


Congruence & Cosets


Three immediate consequences:

(i) If I is the identity of G, then IX D X for all X in G.

) X0 D .IX/0 D I0X0;

for all X0 in G0. Thus, I0 is the identity in G0.

(ii) Further,

I0 D .XX1/0 D X0.X1/0

That is, .X1/0 D .X0/1.

(iii) If element X in G is of order m, i.e. I D Xm, then

I0 D .Xm/0 D .XXm1/0

D X0.Xm/0 D D X0X0 : : : X 0

m factors

Two

This third consequence should not be here.



50/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

yMapping

yMapping 2

yMapping 3

yMapping 4

yMapping 5

Subgroups


Congruence & Cosets


The image of an element has the same order as the element.

What distinguishes an isomorphism from the more general

homomorphism are the requirements that in an isomorphism:

(I) different elements in G must map into different elements in G0

(whereas in a homomorphism several elements in G may

have the same image in G0), that is, x0 D y0 must imply

x D y;

(II) any element in G0 must be the image of some element in G.

Isomorphism.



51/78


Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

yMapping

yMapping 2

yMapping 3

yMapping 4

yMapping 5

Subgroups


Congruence & Cosets


For a homomorphism, the set of elements of G whose image in G0

is I0

iscalled the kernel of the homomorphism. In an isomorphism the kernel

consists of the identity I alone.

Example: Consider a mapping between the additive group of realnumbers R and the multiplicative group of complex numbers with unit

modulus, U.1/. Suppose that the mapping R ! U.1/ is

W x ! eix

then this is a homomorphism since

.x C y/0 ! ei.xCy/ D eixeiy D x0y0:

It is not an isomorphism since many (an infinite number) of the elementsof R have the same image in U.1/. Eg., ;3;5;::: in R all have the

image 1 in U.1/. Also, all elements of R of the form 2 n, where n is an

integer, map onto the identity element in U.1/, and therefore form the

kernel of the homomorphism.


52/78

Subgroups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


Subgroups


53/78

Subgroups

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


(a)

I A B C D E

I I A B C D E

A A B I E C D

B B I A D E C

C C D E I A B

D D E C B I A

E E C D A B I

(b)

I A B C

I I A B C

A A I C B

B B C I A

C C B A I

Table. 9: Reproduction of (a) Table 8 and (b) Table 3 with the relevant

subgroups shown in bold.

Tables 3 and 8 show that the upper left corners of each table haveproperties associated with a group multiplication table (see

Table 9) (subgroup).

Subgroups 2


54/78

Subgroups 2

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


A subgroup of a group G is defined as any non-empty subsetH D fHi g of G, the elements of which themselves behave as a

group under the same rule of combination as applies in G itself.

The order of the subgroup (denoted by h or jHj) is equal to the

number of elements.

All groups G contain two trivial subgroups;

(i) G itself,

(ii) the set I consisting of the identity element alone.

All other subgroups are termed proper subgroups. In a group with

multiplication Table 8 the elements fI ; A ; Bg form a proper

subgroup, as do fI; Ag in a group with Table 3 as its group table.

Subgroups 3


55/78

Subgroups 3

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


Some groups have no proper subgroups.

(a)

I A BI I A B

A A B I

B B I A

(b)

I A B C D

I I A B C DA A B C D I

B B C D I A

C C D I A B

D D I A B C

Table. 10: The group tables of two cyclic groups, of orders 3 and 5. They

have no proper subgroups.

Tables 10(a) and (b) show the multiplication tables for two of these

groups.

Subgroups 4


56/78

Subgroups 4

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


The order of any group is a multiple of the order of any of its

subgroups (Lagranges theorem) i.e. in our general notation, g is a

multiple of h. Thus, a group of order p, where p is any prime, must

be cyclic and cannot have any proper subgroups [eg. Table 10(a)and (b)].

Subgroups 5


57/78

Subgroups 5

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


Repeated multiplication of an element X (not the identity) by itselfwill generate a subgroup fX; X2; X3; : : : g. The subgroup will

clearly be Abelian, and if X is of order m i.e. Xm D I, the

subgroup will have m distinct members.

If m is less than g, m must be a factor of g (Lagranges theorem).

Also, the order of any element of a group is an exact divisor of the

order of the group.

Subgroups 6


58/78

Subgroups 6

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


Some properties of the subgroups of a group G are as follows:

(i) The identity element of G belongs to every subgroup H.

(ii) If element X belongs to a subgroup H, so does X1.

(iii) The set of elements in G that belong to every subgroup of G,

themselves form a subgroup, though it may consist of the

identity alone.


59/78

Subgroups 8


60/78

Subgroups 8

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups

ySubgroups

ySubgroups 2

ySubgroups 3

ySubgroups 4

ySubgroups 5

ySubgroups 6

ySubgroups 7

ySubgroups 8


Congruence & Cosets


To prove (i), suppose Z and W belong H0

, with Z D X0

and W D Y0

,where X and Y belong to G. Then

ZW D X0Y0 D . X Y /0

and therefore belongs to H0, and

Z1 D .X0/1 D .X1/0

and therefore belongs to H0. These two results, together with the fact

that I0 belongs to H0, are enough to establish result (i).

To prove (ii), suppose X and Y belong to K, then

. X Y /0

D X0

Y0

D I0

I0

D I0

.closure/

I0 D .XX1/0 D X0.X1/0 D I0.X1/0 D .X1/0

and therefore X1 belongs to K. These two results, together with the

fact that I belongs to K, are enough to establish result (ii).


61/78

Equivalence Relations and Classes

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


ySubdividing a Group

yEquiv Relations

yEquiv Classes

yEquiv Classes 2

Congruence & Cosets


Subdividing a Group


62/78

g p

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups



yEquiv Relations

yEquiv Classes

yEquiv Classes 2

Congruence & Cosets


We will identify ways in which the elements of a group can bedivided up into sets with the property that each element of the

group belongs to one, and only one, such set.

The subgroups of a group clearly do not form such a partitionbecause the identity element is in every subgroup (rather than

being in only one).

Equivalence Relations


63/78

q

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups



yEquiv Relations

yEquiv Classes

yEquiv Classes 2

Congruence & Cosets


An equivalence relation on a set S is a relationship X Ybetween two elements X and Y belonging to S , in which the

definition of must satisfy the requirements of

(i) reflexivity, X X;

(ii) symmetry, X Y implies Y X;

(iii) transitivity, X Y and Y Z imply X Z.

Equivalence Classes


64/78

q

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups



yEquiv Relations

yEquiv Classes

yEquiv Classes 2

Congruence & Cosets


Theorem:

An equivalence relation on S divides up S into classes Ci such

that

(i) X and Y belong to the same class if, and only if, X Y;

(ii) every element W of S belongs to exactly one class.

Equivalence Classes 2


65/78

q

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups



yEquiv Relations

yEquiv Classes

yEquiv Classes 2

Congruence & Cosets


Proof: (() Let X belong to S , and define the subset SX of S to bethe set of all elements U of S such that X U. Clearly by

reflexivity X belongs to SX. Suppose first that X Y, and let Z

be any element of SY. Then Y Z, and hence by transitivity

X Z, which means that Z belongs to SX. Conversely, since thesymmetry law gives Y X, if Z belongs to SX then this implies

that Z belongs to SY. These two results together mean that the

two subsets SX and SY have the same members and hence are

equal.

()) Now suppose that SX equals SY. Since Y belongs to SY it

also belongs to SX and hence X Y. This completes the proof of

(i), once the distinct subsets of type SX are identified as the

classes Ci .

Statement (ii) is an immediate corollary, the class in question being

identified as SW.


66/78

Congruence and Cosets

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Lagranges Theorem


67/78

g g

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Theorem:If G is a finite group of order g, and H is a subgroup of G of order

h, then g is a multiple of h.

Proof:Definition of : given X and Y belonging to G, X Y , X1Y

belongs to H.

ie. Y D XHi for some element Hi belonging to H; X and Y are

said to be left-congruent with respect to H.

Lagranges Theorem 2


68/78

g g

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


This defines an equivalence relation:

(i) Reflexivity: X X, since X1X D I and I belongs to any

subgroup.

(ii) Symmetry: X Y implies that X1Y belongs to H, and so,

therefore, does its inverse, since H is a group. But

.X1Y /1 D Y1X and, as this belongs to H, it follows

that Y X.

(iii) Transitivity: X Y and Y Z imply that X1Y and Y1Z

belong to H, and so therefore does their product

.X1Y /.Y1Z/ D X1Z, from which it follows that X Z.

Lagranges Theorem 3


69/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Since is an equivalence relation, it divides G into disjointclasses, which are called the left cosets of H. Thus each element

of G is in one and only one left coset of H.

The left coset containing any particular X is written XH; it mustcontain h different elements, since if it did not, and two elements

were equal,

XHi D XHj;

we have Hi D Hj and H contained fewer than h elements.

Thus the left cosets of H are a partition of G into a number of

sets each containing h members. Since there are g members of G,

and each must be in just one of the sets, it follows that g is a

multiple of h.

Lagranges Theorem 4


70/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


The number of left cosets of H in G is known as the index of H inG and written G W H; numerically the index D g= h.

Lagranges theorem says that any group of order p, where p is a

prime, must be a cyclic and cannot have any proper subgroups:since any subgroup must have an order that divides p, this can

only be 1 or p.

Example 6


71/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


ExampleFind the left cosets of the proper subgroup H of the group G that

has Table 8 as its multiplication table.

SolutionThe subgroup consists of the set of elements H D fI ; A ; Bg. It

has order 3, which is a divisor of 6, the order of the group. H itself

provides the first (left) coset:

IH D fII;IA;IBg D fI ; A ; Bg:

We continue by choosing an element not already selected, C say,

and formCH D fCI;CA;CBg D fC ; D ; Eg:

These two cosets of H exhaust G, and are therefore the only

cosets, the index of H in G being equal to 2.

Example 6 contd


72/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Note that it would not have mattered if we had taken D, say,instead of I to form a first coset

DH D fDI; DA; DBg D fD ; E ; C g

and then, from previously unselected elements, picked B , say:

BH D fBI;BA;BBg D fB ; I ; Ag

The same two cosets would have resulted.

Conjugate and Classes


73/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Definition of : given X and Y belonging to G,X Y , Y D G1i XGi , where Gi is an (appropriate) element of

G.

Different pairs of elements X and Y will, in general, requiredifferent group elements Gi . Elements connected in this way are

said to be conjugates.

Conjugate and Classes 2


74/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Proof:

(i) Reflexivity: X X, since X D I1XI and I belongs to the

group.

(ii) Symmetry: X Y implies Y D G1i XGi and therefore

X D .G1i /Y G1i . Since Gi belongs to G, so does G

1i ,

and it follows that Y X.

(iii) Transitivity: X Y and Y Z imply Y D G1i XGi and

Z D G1j Y Gj and therefore

Z D G1j G1i XGi Gj D .Gi Gj/

1X.Gi Gj/. Since Gi and

Gj belong toG

so does Gi Gj, from which it follows thatX Z.

These results establish conjugacy as an equivalence relation, and

hence show that it dividesG

into classes.



75/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Immediate corollaries are:

(i) If Z is in the class containing I, then

Z D G1

i

IGi D G1

i

Gi D I:

Thus, since any conjugate of I can be shown to be I, the

identity must be in a class by itself.

(ii) If X is in a class by itself, then Y D G1i XGi must imply thatY D X. But

X D Gi G1i XGi G

1i

for any Gi and so

X D Gi .G1i XGi /G

1i D Gi Y G

1i D Gi XG

1i

i.e. XGi D Gi X for all Gi .



76/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


Thus commutation with all elements of the group is anecessary (and sufficient) condition for any particular group

element to be in a class by itself. In an Abelian group each

element is in a class by itself.

(iii) In any group G the set S of elements in classes by

themselves is an Abelian subgroup (known as the centre of

G). We have shown that I belongs to S , and so if, further,

XGi D Gi X and Y Gi D Gi Y for all Gi belonging to G, then

(a) .XY /Gi D XGi Y D Gi . X Y / i.e. the closure of S , and

(b) XGi D Gi X implies X1Gi D Gi X

1, i.e. the inverse

of X belongs to S .

Hence S is a group, and clearly Abelian.

Example 7


77/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd


ExampleFind the conjugacy classes of the group G having Table 8 as its

multiplication table.

SolutionI is in a class by itself. Consider next the results of forming

X1AX, as X runs through the elements of G.

I1

AI D IA D A; A1

AA D IA D AB1AB D AI D A; C1AC D CE D B

D1AD D DC D B; E1AE D ED D B

Only A and B are generated. It is clear that fA; Bg is one of theconjugacy classes of G. This can be verified by forming all

elements X1BX; again only A and B appear.

Example 7 contd


78/78

Group Theory

Cyclic Group

Finite Groups

Non-Abelian Groups

Permutation Groups

Mapping

Subgroups


Congruence & Cosets

yLagrange

yLagrange 2

yLagrange 3

yLagrange 4

yExample 6

yExample 6 contd

yConjugacy Class

yConjugacy Class 2

yConjugacy Class 3

yConjugacy Class 4

yExample 7

yExample 7 contd

We now need to pick an element not in the two classes alreadyfound. Suppose we pick C. We compute X1CX, as X runs

through the elements of G. The calculations give the following.

X W I A B C D EX1CX W C E D C E D

Thus C, D, and E belong to the same class. The group is now

exhausted, and so the three conjugacy classes are

fIg; fA; Bg; fC ; D ; Eg:

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5 Group Theory