Homomorphisms, Normal Subgroup & Conjugation

Homomorphisms

Group Homomorphism

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

Proposition

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

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 and conjugate if there exists such that .

Normal Subgroup

Let be a group. A subgroup is called normal if for every . We denote a normal subgroup by .

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

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.

Definition

Def Simple Group A group is simple if there’s no non-trivial normal subgroups.

Normaliser

Let be a group and . The normaliser is the subgroup of that In other words, it is the largest subgroup of in which is normal.

Proposition

.

Isomorphisms

Group Isomorphism

A group homomorphism is called isomorphism if it has an inverse, that is a homomorphism such that and .

Proposition

Prop A homomorphism is an isomorphism if and only if it is bijective.

Proposition

Prop Groups of prime order are cyclic.

Proposition

Prop Any two cyclic group of the same order are isomorphic.

Definition

Def Lambda-Map For every define a map by left multiplication with :

Proposition

Prop .

Proposition

Prop is the inverse of .

Cayley’s Theorem

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

Corollary Every finite group is isomorphic to a subgroup of a symmetric group . Proof For any finite group , the map forms an isomorphism, thus for some .

Definition

Def Automorphism An automorphism is an isomorphism from a group to . Denote as the set of all automorphisms on .

Conjugation

Definition

Def Conjugation Let be a group and . The mapis called conjugation by . We say that the elements and are conjugate.

Proposition

Prop Let be a group and . Then is an isomorphism. Proof We first check that is a homomorphism:The map is injective because . And is surjective since for all , we have .