Question

Define a Normal subgroup of a group and give an example.

Answer 1

Answer:

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

A subgroup N of a group G is known as normal subgroup of G if every left coset of N in G is equal to the corresponding right coset of N in G. That is, gN=Ng for every g ∈ G . A subgroup N of a group G is known as normal subgroup of G, if h ∈ N then for every a ∈ G aha-1 ∈ G .

Example :

Let G be a group and let H be a subgroup of G. We have already proven the following equivalences:

1) H is a normal subgroup of G.

2) gHg−1⊆H for all g∈G.

3) NG(H)=G.

4) There exists a homomorphism φ on G such that H=ker(φ).

We will now look at some examples of normal subgroups of groups.

Example 1

Let φ:G→G by the identity isomorphism defined for all g∈G by φ(g)=g. Since φ is an isomorphism we have that φ is injective, and so ker(φ)={1}, the trivial group. Let H={1}. Then by (4) we have that the trivial subgroup is always a normal subgroup of any group G.