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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.
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