GFaktor From Hom

8/6/2019 GFaktor From Hom

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Introduction

Let Gb a group and let Hbe asubgroup of G.

If H is the kernel of a group

homomorphisms G Gthen the left cosets of H will be

elements of a group whose binaryoperation is derived from the

group operation of G.


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Introduction (cont.)

+ 0 3 1 4 2 5

0 0 3 1 4 2 5

3 3 0 4 1 5 2

1 1 4 2 5 3 0

4 4 1 5 2 0 3

2 2 5 3 0 4 1

5 5 2 0 3 1 4

K B M

K K B M

B B M K

M M K B

See Tables


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Introduction (Cont.)

Let G G be a grouphomomorphisms with kernelH.

For a G,

is the left coset aHofHand is also the rightcosetHa ofH.

1 |a x G x a


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Since these left and right cosets ofHcoincide,we will simply refer to them as cosets ofH.

Now is a group (Property ofHomomorphisms). We associate with each

the coset .

By renaming by the name of theassociate coset, that is, by , we canconsider the cosets to form a group.

G

y G 1

| y x G x a

y G 1 y


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This group will be isomorphic to since itis just renamed.

In summary, the cosets of the kernel of agroup homomorphisms form a group

isomorphic to the subgroup of .

The binary operation on the cosets can be

computed in terms of the group operation of .

G

: 'G G

G

'G

G

'G

This group of cosets is the factor group of Gmodulo H, and is denoted by G/H.


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Theorem

Let be a group homomorphisms

with kernel H. Then the cosets of H form agroup, G/H, whose binary operationdefines the product (aH)(bH) of two cosets

by choosing elements a and b from thecosets, and letting

(aH)(bH) = (ab)H.

Also, the map defined by

is an isomorphisms.

: 'G G

: /G H G aH a


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Proof

The first thing we have to worry about isthe definition (aH)(bH) = (ab)H for the

product of two cosets in G/H.

The product is compued by choosing anlement from each of the cosets aHand bH,

and b finding the coset containing theirproduct ab.


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Proof (cont.)

Any time the definition of something involvesmaking choices, we should show that the end

result is independent of the choices made.

We say that a thing is well defined if it isindependent of any choices made in its

computation.


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Proof (cont.)

Thus we start by showing that

(aH)(bH) = (ab)H

gives a well-defined operation on G/H.

Suppose that and

are two other representative elements from

these cosets.

1ah aH

2bh bH


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Proof (cont.)

We must show that

that is, that ab and lie in the same cosetof G/H.

We need only show that

1 2 .ab ah bh

1 2ab H ah bh H

1 2ahbh

Recall that for all 'h e h H


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Proof (cont.)

Thus

.ab

1 2 1 2ah bh a h b h

Thus we do indeed have a well-definedbinary operation on G/H.

' '

a e b e

a b


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Proof (cont.)

We now check the group axioms for G/H.

aH bH cH aH bc H a b c H

Associative

ab cH

ab H cH

.aH bH cH


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Proof (cont.)

aH eH ae H

aH

Indentity

ea H

.eH aH

SoH= eHacts as indentity coset in G/H.


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Proof (cont.)

1 1a H aH a aH eH

Inverses

1aa H

1.aH a H

So is the inverse of aHin G/H.1

a H


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Proof (cont.)

aH bH ab H ab

Homomorphisms

a b

.aH bH


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Proof (cont.)

One to one

a b aH bH Suppose . Then ,

so that aH and bH are the same coset.

Let Then for someand

Onto

.y G

y x

x G

. xH x y


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Example

Considered the map 4:

where is the remainder when m is

devided by 4 in accordance with the division

algoritm.

m

We know that is a homomorphisms.

ker 4 Of courses


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Example (cont.)

Considered the map

4 , 8, 4,0, 4,8,

We know that is aisomorphisms.

4: / 4

1 4 , 7, 3,1,5,9, 2 4 , 6, 2, 2,6,10,

3 4 , 5, 1,3,7,11,


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Example (cont.)

4: / 4

4 0

1 4 1

2 4 2

3 4 3

GFaktor From Hom

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