1

1

Group

Chapter 2

2

Binary operation in setLet be a set, a binary operation is a mapping ,which means that for any two elements we can find such that according to this mappiNote: the operation result eleme

, ng.

t u n m

G G G G

x y G z G

z x y

z

× →∈ ∈

=

����

����

����

Let be the set of all integers. We define the binary operation to be regular addition +.If we pick = 2 and =3, Then

st be in (closure property)

2 3 5

Example 1.

Example 2. Let be the

G

Z

Z

x y x y x y= + = + =����

set of all integers. We define the binary operation to be regular multiplication *.If we pick = 2 and =3, Then * 2*3 6x y x y x y= = =����

2

3

Examples2

2

. Let be the set of all 2 2 matrices in real numbers.

Define the operation to be the ma

Ex

trix addition +. Then

1 2 3 7 1 2 3 7 4 93 4 1 2 3 4 1 2 2 6

.

ample 3

LetExampl 4 be e

M

M

×

� � � � � � � � � �= + =� � � � � � � � � �− −� � � � � � � � � �

����

����

the set of all 2 2 matrices in real numbers.

Define the operation to be the matrix multiplication *. Then

1 2 3 7 1 2 3 7*

3 4 1 2 3 4 1 2

3 2 7 4 1 119 4 21 8 5 29

×

� � � � � � � �=� � � � � � � �− −� � � � � � � �

− +� � � �= =� � � �− +� � � �

����

����

4

Definition of GroupLet be an non-empty set defined a binary operation . If 1)

2) There exists element such that and .such is called the identity elem

(associative)

ent (

, , ,

f

( ( )

oe

)G

a b c G a b c a b c

e a G e a a a e ae G

∀ ∈ =

∀ ∈ = =

����

� � � �� � � �� � � �� � � �

� �� �� �� �

1

3) there exists element such that and .

such is called the inverse element of de

xistence of identity element)

(existence of inverse elementnoted as

Then we say tha

,

( ,t is)

) a gro

a G b b a e a b e

b a a

G

−

∀ ∈ = =� �� �� �� �

���� up. If there is no confusion about the operation, we simply say that is a group. Also we can omitnotation to write as if without confusion

G

a b ab� �� �� �� �

3

5

1. In the above example, ( Z, + )is a group whose identityelement is zero and the inverse element of integer n is the

opposite integer −n2. ( Z, * ) is not a group. Although it has identity element,which is 1. But some elements like 0, 3 have no inverse in Z

( You may think that 3−1 is the inverse of 3, however, 3−1∉ Z)

2

2

2

0 00 0

1 0.

0 1

3. is a group. Identity element is

If , its inverse is

4. is a not a group. Alth

( +)

(

ough it has identity elemen

+)

( *) t matrix

This is b

a b a bc d c d

e

M

M

M

� �� �� �

� � � �− −� � � �− −� � � �

� �= � �� �

∈

ecause, if the determinant of a matrix is zero,

then it has no inverse matrix.

6

11

11

1det( )

1 23 4

1 2 4123 4

The determinant of matrix is value

If then ha

det( )

det( ) 0

det( ) 1

s inverse

Fro example, then 4 2 3 2

S

0

o

a b

c d

a b d bMc d c a

M M ad bc

M M

M

M M

M

−−

−−

� �� �� �

� � � �−� � � �−� � � �

� �� �� �

� � −� � −� �

= = −

≠

= =

= = × − × = − ≠

= = 2 2 13 1 1.5 0.5

1 2 2 1 2 3 1 1 1 0*

3 4 1.5 0.5 6 6 3 2 0 1 Checking:

Note. If then we say matrix is singular. Otherwiswe

we say is none-singul

det( ) 0,

ar.

M M

M

� � � �−=� � � �− −� � � �

� � � � � � � �− − + −= =� � � � � � � �− − −� � � � � � � �

=

4

7

ExamplesExample 5: Let be the set of all 2×2 non-singular matrices in realnumbers. Define the operation to be the matrix multiplication *.

Then (G, * ) is a group.

Example 6: Let R+ be the set of all positive real numbers, then (R+, * ) is a group, with identity =1.

Example7: Let R+ be the set of all positive real numbers, then (R+, + ) is a not group because it has no identity element. (for real number and addition operator, the identity element is zero, but 0 ∉ R+ )

8

Examples continuedExample 8: Let G = {1} then (G, *) is a group, here 1 is identity element and its inverse is 1 too.Note: In a group, it is always true that e−1=e

Example 9: Let G = {1, −1} , then (G, *) is a group, here the inverse of −1 is −1 too.

Example 10: Let G = {1, −1, i, −i} , then (G, *) is a group, herethe inverse of i is −i .

{ }1 3 1 32 2 2 2

1 3 1 32 2 2 2

: Let then ( ,*}is a group,

here and are inverse of eac

Example

h othe

11 ,

r.

1,G i i

i i

G+ −

+ −

= − −

− −

5

9

Some properties of GroupProperty 1: Let G be a group, then G has unique identity elementProof: If G has two identity elements e1 and e2 then

e1 =e1e2 =e2

Property 2: Let G be a group, then every element of G has only one inverse elementProof: If a ∈ G has two inverse element b1 and b2 then

ab1 = b1a =e and ab2 = b2a =eSo b1 = b1e= b1(a b2) = (b1a)b2 = eb2 = b2

Property 3: Let G be a group, then ∀a, b, c ∈G, ab = ac � a−1ab = a−1ac � b = c

and ba = ca � baa−1 = caa−1 � b = c

10

Group may not be commutativeDefinition: Let G be a group, if ∀a, b∈G,

ab = baWe say that G is a commutative group.

However, not all groups are commutative. For example, matrix multiplication operation is not commutative.

1 2 3 7 3 2 7 4 1 113 4 1 2 9 4 21 8 5 29

3 7 1 2 3 21 6 28 24 341 2 3 4 1 6 2 8 5 6

− +� � � � � � � �= =� �� � � � � �− − +� �� � � � � �

+ +� � � � � � � �= =� �� � � � � �− − + − +� �� � � � � �

So the group of all 2×2 non-singular matrices is not commutative.