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.