correspondence between normal subgroups and homomorphic images

correspondence between normal subgroups and homomorphic images

Assume, that

G

and

H

are groups. If

f

:

G

→

H

is a

group homomorphism

, then the

first isomorphism theorem

states, that the

function

F

:

G

/

ker

⁢

(

f

)

→

im

⁢

(

f

)

defined by

F

⁢

(

g

⁢

ker

⁢

(

f

)

)

=

f

⁢

(

g

)

is a well-defined group isomorphism. Note that

ker

⁢

(

f

)

is always normal in

G

.

This leads to the following question: is there a correspondence between

normal subgroups

of

G

and

homomorphic images

of

G

? We will try to answer this question, but before that, let us introduce some notion.

First of all, homomorphic image

im

⁢

(

f

)

is not only a

subgroup

of

H

. Actually homomorphic image contains also some data about

homomorphism

. This observation leads to the following definition:

Definition.

Let

G

be a group. Pair

(

H

,

f

)

is called

a homomorphic image of

G

iff

H

is a group and

f

:

G

→

H

is a

surjective

group homomorphism. We will say that two homomorphic images

(

H

,

f

)

and

(

H

′

,

f

′

)

of

G

are

isomorphic

(or

equivalent

), if there exists a group isomorphism

F

:

H

→

H

′

such that

F

∘

f

=

f

′

.

It is easy to see, that this

isomorphism

relation

is actually an

equivalence relation

and thus we may speak about isomorphism classes of homomorphic images (which will be denoted by

[

H

,

f

]

for homomorphic image

(

H

,

f

)

). Furthermore, if

N

⊂

G

is a normal subgroup, then

(

G

/

N

,

π

N

)

is a homomorphic image, where

π

N

:

G

→

G

/

N

is a

projection

, i.e.

π

N

⁢

(

g

)

=

g

⁢

N

. Let

norm

⁢

(

G

)

=

{

N

⊆

G

|

N

⁢

is normal subgroup

}

;

h

.

im

⁢

(

G

)

=

{

[

H

,

f

]

|

(

H

,

f

)

⁢

is a homomorphic image of

⁢

G

}

.

Proposition

.

Function

T

:

norm

(

G

)

→

h

.

im

(

G

)

defined by

T

⁢

(

N

)

=

[

G

/

N

,

π

N

]

is a

bijection

.

Proof.

First, we will show, that

T

is onto. Let

(

H

,

f

)

be a homomorphic image of

G

. Let

N

=

ker

⁢

(

f

)

. Then (due to the first isomorphism theorem), there exists a group isomorphism

F

:

G

/

N

→

H

defined by

F

⁢

(

g

⁢

N

)

=

f

⁢

(

g

)

. This shows, that

f

⁢

(

g

)

=

F

⁢

(

g

⁢

N

)

=

F

⁢

(

π

N

⁢

(

g

)

)

=

(

F

∘

π

N

)

⁢

(

g

)

and thus

(

G

/

N

,

π

N

)

is isomorphic to

(

H

,

f

)

. Therefore

T

⁢

(

N

)

=

[

G

/

N

,

π

N

]

=

[

H

,

f

]

,

which

completes

this part.

Now assume, that

T

⁢

(

N

)

=

T

⁢

(

N

′

)

for some normal subgroups

N

,

N

′

∈

norm

⁢

(

G

)

. This means, that

(

G

/

N

,

π

N

)

and

(

G

/

N

′

,

π

N

′

)

are isomorphic, i.e. there exists a group isomorphism

F

:

G

/

N

→

G

/

N

′

such that

F

∘

π

N

=

π

N

′

. Let

x

∈

N

′

=

ker

⁢

(

π

N

′

)

and denote by

e

∈

G

/

N

′

the

neutral element

. Then, we have

e

=

π

N

′

⁢

(

x

)

=

F

⁢

(

π

N

⁢

(

x

)

)

and (since

F

is an isomorphism) this is if and only if

x

∈

ker

⁡

(

π

N

)

=

N

. Thus, we’ve shown that

N

′

⊆

N

. Analogously (after considering

F

-

1

) we have that

N

⊆

N

′

. Therefore

N

=

N

′

, which shows, that

T

is

injective

. This completes the proof.

□

Title

correspondence between normal subgroups and homomorphic images

Canonical name

CorrespondenceBetweenNormalSubgroupsAndHomomorphicImages

Date of creation

2013-03-22 19:07:11

Last modified on

2013-03-22 19:07:11

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Theorem

Classification

msc 20A05

Classification

msc 13A15

Related topic

HomomorphicImageOfGroup

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