Section 13 Homomorphisms
Definition
A map of a group G into a group G’ is a homomorphism if the homomophism property
(ab) = (a)(b)
Holds for all a, bG.
Note: The above equation gives a relation between the two group structures G and G’.
Example:
For any groups G and G’, there is always at least one homomorphism: : G G’ defined by (g)=e’ for all g G, where e’ is the identity in G’. We call it the trivial homomorphism.
Examples
Example
Let r Z and let r: Z Z be defined by r (n)=rn for all n Z. Is r a homomorphism?
Solution: For all m, n Z, we have r(m + n) = r(m + n) = rm + rn = r (m)+ r (n). So r is a homomorphism.
Example:
Let : Z2 Z4 Z2 be defined by (x, y)=x for all x Z2, y Z4. Is a homomorphism?
Solution: we can check that for all (x1, y1), (x2, y2) Z2 Z4,
((x1, y1)+(x2, y2) )= x1+ x2= (x1, y1)+ (x2, y2).
So is a homomorphism.
Composition of group homomorphisms
In fact, composition of group homomorphisms is again a group homomorphism.
That is, if : G G’ and : G’ G’’ are both group homomorphisms then their composition ( ): G G’’, where
( )(g)= ((g)) for g G. is also a homomorphism.
Proof: Exercise 49.
Properties of Homomorphisms
Definition
Let be a mapping of a set X into a set Y, and let AX and B Y.
• The image [A] of A in Y under is {(a) | a A}. • The set [X] is the range of . • The inverse image of -1 [B] of B in X is {x X | (x) B }.
Theorem
Theorem
Let be a homomorphism of a group G into a group G’.
1. If e is the identity element in G, then (e) is the identity element e’ in G’.
2. If a G, then (a-1)= (a)-1.
3. If H is a subgroup of G, then [H] is a subgroup of G’.
4. If K’ is a subgroup of G’, then -1 [K’] is a subgroup of G.
Proof of the statement 3:
Let H be a subgroup of G, and let (a) and (b) be any two elements in [H]. Then (a) (b)= (ab), so we see that (a) (b) [H]; thus [H] is closed under the operation of G’. The fact that e’= (e) and (a-1)= (a)-1 completes the proof that [H] is a subgroup of G’.
Kernel Collapsing
Definition
Let : G G’ be a homomorphism of groups.
The subgroup -1[{e’}]={x G | (x)=e’} is the kernel of , denoted by Ker().
Let H= Ker() for a homomorphism . We think of as “collapsing” H down onto e’.
G’
G
a’ (b) e’ (x) y’
-1[{a’}] bH H Hx -1[{y’}]
be
x
Theorem
Theorem
Let : G G’ be a group homomorphism, and let H=Ker(). Let a G. Then the set
-1[{(a)}]={ x G | (x)= (a)}
is the left coset aH of H, and is also the right coset Ha of H.
Consequently, the two partitions of G into left cosets and into right cosets of H are the same.
Corollary
A group homomorphism : G G’ is a one-to-one map if and only if Ker()={e}.
Proof. Exercise.
Normal Subgroup
Definition
A subgroup H of a group G is normal if its left and right cosets coincide, that is if gH = Hg for all gG.
Note that all subgroups of abelian groups are normal.
Corollary
If : G G’ is a group homomorphism, then Ker() is a normal subgroup of G.
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