homo morph ism

8/6/2019 Homo Morph Ism

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DefinitionSuppose (G,*) and (G,) be a group. A map G G is a homomorphism if

(a*b) = (a) (b)

for all a,b G.

For any groups G and G, there is always at least

one homomorphism G G, namely thetrivial homomorphism defined by (a) = e for alla G, where e is the identity in G.


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Example Example 1 Let G G be a group homomorphism

of G onto G. We claim that if G is abelian, then G mustbe abelian.

Solution Let a,b G. We must show that ab= ba.

Since is onto G, there exist a,b G such that (a) =aand (b) =b. So

ab= (a)(b)

=(ab) (Since is homomorphism)

=(ba) (Since G is abelian, we have ab =ba.)= (b)(a) (Since is homomorphism)

= ba. (Since is onto G)


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Example

Example 2 Let Sn be the symmetric group on nletters, and let Sn Z2 be defined by

Show that is a homomorphism.

n.permutatiooddanisif1n,permutatioevenanisif0


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Example

Example 3 Let F be the additive group ofall functions mapping R into R, let R be the

additive group of real numbers, and c beany real number. Let c F R be theevaluation homomorphism defined by for g

F. Recall that, by definition, the sum oftwo functions g and h is the function g + hwhose value at x is g(x) + h(x). Thus we

have


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Properties of Homomorphism

Definition Let be a mapping of a set Xinto a set Y, and A X and B Y. The

image [A] of A in Y under is{(a) | aA}.

The set [X] is sometimes called the

range of . The invers image -1[B] of B inX is

{xX | (x) B}


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Theorem

Let be a homomorphism of a group Ginto a group G.

1. If e is the identity in G, then (e) is theidentity e in G.

2. If a G then (a-1) = (a)-1.

3. If H G, then [H] G.4. If K G, then -1[K] G.


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Definition

Let G G be ahomomorphism of groups. Thesubgroup

-1[e] = {x G| (x) = e}

is the kernel of , denoted byKer().


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Theorem

Let G G be a group homomorphism,

and letH= Ker(). Let a G, then the set

-1[(a)] = {x G| (x) = (a)}is the left coset aHofH, and is also the rightcosetHa ofH.

Consequently, the two partitions of G into leftcosets and into right cosets ofHare the same.


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Proof

We want to show that

-1[(a)] = {x G| (x) = (a)} = aH.

Suppose that (x) = (a). Then(a)-1(x) = e

where eis the identity of G. By Theorem

we know that (a)-1 = (a-1), so we have(a-1)(x) = e.


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Proof (Cont.)

Since is a homomorphism, we have

(a-1)(x) = (a-1x) = e.

So (a-1x) = e.This show that a-1x H= Ker(), so a-1x=hfor some h H, andx = ah aH.

This show that-1[(a)] = {x G| (x) = (a)} aH.


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Example

Let Dbe the additive group of alldifferentiable functions mapping R into R,

and let F be the additive group of allfunctions mapping R into R.

Then differentiation gives us a map D F, where (f) =f forfD.

We easily see that is a homomorphism,for (f + g) = (f + g)=f+ g )=(f) + (g).


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Example (Cont.)

Now Ker()= {fD | (f) =f= 0}.

Thus Ker() consists of allconstant functions.


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Corollary

A group homomorphism

G G is a one-to-onemap iff

Ker() = {e}.


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Proof

If Ker() = {e}, then for every

a G,the elements mapped into(a) are precisely the element of

the left coset a{e} = {a}, which

show that is one to one.


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Proof (Cont.)

Conversely, suppose is one to

one. We know that (e) = e, the

identity of G. Since is one to one,we see that e is the only elementmapped into e by , so Ker() = {e}.


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Definition

A subgroup H of a group G is

normal if its left and right cosetscoincide, that is, if aH = Hafor all a G.

Note that all subgroups of abeliangroups are normal.


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Note

We mention in closing some terminology foundin the literature related to Ker () and to [G].

A map A B that is one to one is oftencalled an injection.

A homomorphism G G that is one toone is often called a monomorphism.

A map of A onto B is often called a surjection.

A homomorphism that maps G onto G is oftencalled an epimorphism,this is the case iff [G] = G.

homo morph ism

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