Homomorphisms

Definition: Homomorphism

Let and be Monoids Monoids with units , respectively. A map is a monoid homomorphism if and for all . If are Groups Groups, we call a group homomorphism.

Theorem

Let be a map between groups. If for all , is a group homomorphism, and .

Proof:
Since are units, we have so is a group homomorphism. Furthermore, we have so .

Definition

Let be groups. We define...

as the set of all homomorphisms from to ,
as the set of all endomorphisms of ,
as the set of all automorphisms of .

If a group homomorphism is...

Injective, it is called a monomorphism
Surjective, it is called a epimorphism
Bijective, it is called an isomorphism

Theorem: Compositions of homomorphisms are homomorphisms

Let be groups, and . Then Furthermore, if is a group, is a group.
Proof:
We have so . Furthermore, we know that the composition of two bijections is a bijection, hence for all . We also know that the identity bijection is in , and that every bijection has an inverse also in , so is indeed a group.

Definition

Let be groups, and . We define the kernel We also define the image

Theorem

Let be groups, and . Then

is injective
is surjective

Proof:
7. {a} If , then , so which means according to Theorem Theorem.
8. For any we have , meaning .
9. We know , so if is injective we get . If instead and we get so is injective.
10. This is true by definition.
11. For all we get since is a group, so .
12. We have . For any we have since is a group.

Example

Let be a group and be a homomorphism. Then we have So any homomorphism from to a group is uniquely determine by where it maps . Note that this also holds for negative numbers since respects inverses.

Definition: Translation

Let be a group and . We define the (left) translation map by as where for all .

We note that translation by is generally not a homomorphism since . However, the map that takes to is an injective group homomorphism since and every if since and . This means can be identified with its image in , which thus is a subgroup of (for finite groups, this is the result of Cayley's theorem).

Of course, we can also define the right translation map, and get basically the same results, but the maps are of course generally not the same. This asymmetry leads us to also look at conjugation.

Definition Conjugation

Let be a group and . We define the conjugation by as where
for all . is also called an inner isomorphism of . And we define .

We know that because , and is clearly an inverse of . This time, however, we only get that the map that takes to is still a homomorphism, but generally no longer an injective one. It's a homomorphism since and , but generally not injective since for example, if is abelian, we get for every .