Homomorphisms, Normal Subgroup & Conjugation

Homomorphisms

Group Homomorphism

Let and be groups. A group homomorphism is a map that satisfies for all .

Proposition

If is a group homomorphism, then the identity of is mapped to the identity of .

Proof Let and be the identity elements of and , respectively. Then we have . Multiplying both sides by , we get .

Kernel & Image

Let be a group homomorphism. The kernel and the image of are defined by

Proposition

A homomorphism is injective if and only if .

Proof If is injective then clearly . Conversely, suppose that and let with . Then . Hence and therefore .

Normal Subgroup

Conjugate

Let be a group. For any , we say and are conjugate if there exists such that .

Normal Subgroup & Simple Group

Let be a group. A subgroup is called normal if for every . We denote a normal subgroup by .
A group is simple if there’s no non-trivial normal subgroups.

e.g. Let be a homomorphism, then . Fix some , then for all we have .

From the definition of a normal subgroup, we immediately have the following properties:

Proposition

The followings are true:

1. is normal iff the right coset coincides with the left coset for every .
2. We required in the definition that , but it follows that .
3. Every subgroup of an abelian group is normal.
   

It is clear that not every subgroup is normal. For example, let and , then . However, we can make an arbitrary subgroup as normal as possible by two ways. One is to instead of sticking to , find the largest subgroup of in which is normal; the other is to fix , but to find the largest normal subgroup of contained in . The first one is called the normaliser of in , and the second one is called the normal core of in .

Normaliser

Let be a group and . The normaliser is the subgroup of that

Proposition

is the largest subgroup of in which is normal.

Proof We first show that . For all , and , we have so . Now let such that . Then for all , we have , so . Thus , and the result follows.

Normal Core

Let be a group and . The normal core of in is

Proposition

is the largest normal subgroup of contained in .

Proof We first show that . For all , we have , for some and any . Then so .
It is clear that . Now let such that . Then for all , we have for some , so . Thus , and the result follows.

Proposition

Normal core is the kernel of the left multiplication action of on the left coset space .

Proof Recall the left multiplication action of on the left coset space is given by Then the kernel of is which is exactly the normal core of in .

Corollary

is normal in if and only if , if and only if .

Proposition

Suppose , and , then commutes with . Moreover, the can be embedded into by .

Proof For all and , we have since , and since . Therefore, , so which implies that . Then the “moreover” part is easy to verify.

Isomorphisms

Group Isomorphism

A group homomorphism is called an isomorphism if it has an inverse, that is a homomorphism such that and . Equivalently, a homomorphism is an isomorphism if and only if it is bijective.
An automorphism is an isomorphism from a group to . We denote as the set of all automorphisms on .

e.g.

• Any two cyclic group of the same order are isomorphic.
• Let be a group and . The map called conjugation by , is an isomorphism. We first check that is a homomorphism:The map is injective because . And is surjective since for all , we have .
  

Cayley’s Theorem

For every define a map by left multiplication with :
Then , is the inverse of .
Moreover, the map given by is an injective group homomorphism.

Proof Clearly . Set where is the identity map from to . Then for all , we have . It follows that . Thus is injective.

Corollary

Every finite group is isomorphic to a subgroup of a symmetric group .

Proof For any finite group , is moreover bijective.