Below is a clear, simple, and exam-oriented explanation of Normal Subgroups — perfect for BCA/MCA/Engineering/Discrete Mathematics and Abstract Algebra exams.

⭐ NORMAL SUBGROUPS – INTRODUCTION

A normal subgroup is a special type of subgroup in a group.
Normal subgroups allow us to form quotient groups, which are essential for:

✔ Simplifying group structure
✔ Understanding symmetry
✔ Fundamental Theorem of Homomorphisms
✔ Group classification

⭐ 1. Definition of a Normal Subgroup

Let ( G ) be a group and ( N \le G ) a subgroup.

N is a normal subgroup if:

[
gN = Ng \qquad \forall g \in G
]

That is, left and right cosets coincide.

⭐ Equivalent Definitions

The following are equivalent (all mean N is normal):

✔ 1️⃣ Coset Definition

[
gN = Ng \quad \forall g \in G
]

✔ 2️⃣ Conjugation Definition

[
gNg^{-1} = N \quad \forall g \in G
]

OR, for each (n \in N):

[
gn g^{-1} \in N
]

This means N is closed under conjugation.

✔ 3️⃣ Homomorphism Kernel Definition

If ( f : G \to H ) is a homomorphism, then:

[
\ker(f) \triangleleft G
]

Every kernel of a homomorphism is automatically normal.

✔ 4️⃣ Inner Structure

Normal subgroup N is invariant under the action of G.

⭐ Notation

Normal subgroup is denoted by:

[
N \triangleleft G
]

Subgroup:

[
N \le G
]

⭐ 2. Why Normal Subgroups Are Important?

Because only normal subgroups can form quotient groups.

If N is normal:

[
G/N = {gN : g \in G}
]

is a group under coset multiplication:

[
(aN)(bN) = (ab)N
]

If N is not normal → quotient is NOT a group.

⭐ 3. Examples of Normal Subgroups

✔ Example 1: Center of a group

[
Z(G) = {g \in G : gx = xg \ \forall x \in G}
]

Always normal.

✔ Example 2: Kernel of a homomorphism

Let
[
f:G \to H
]
Then:

[
\ker(f) = {g \in G : f(g)=e_H}
]

always normal.

✔ Example 3: All subgroups in Abelian groups are normal**

If G is abelian:

[
gx = xg
]

So for any G abelian:

[
H \le G \Rightarrow H \triangleleft G
]

Example:

((\mathbb{Z},+))
((\mathbb{R},+))

Every subgroup is normal.

✔ Example 4: The subgroup ( A_n ) inside ( S_n )

(S_n) = symmetric group
(A_n) = alternating group (even permutations)

[
A_n \triangleleft S_n
]

⭐ 3. Examples of Subgroups that Are NOT Normal

✘ Example 1: Subgroup of upper triangular matrices in (GL(2,\mathbb{R}))

Not invariant under conjugation.

✘ Example 2: In (S_3)

Let:
[
H = { e, (12) }
]

Check conjugation:

[
(13)(12)(31) = (23) \notin H
]

Thus:
[
H \not\triangleleft S_3
]

⭐ 4. Tests for Normality

✔ Test 1: Coset test

N is normal if:

[
gN = Ng \quad \forall g \in G
]

✔ Test 2: Conjugation test

N is normal if:

[
gng^{-1} \in N \quad \forall g \in G,\ n \in N
]

✔ Test 3: Index 2 test

If N has index 2 in G:

[
[G : N] = 2
]

→ N is always normal.

Reason: only two cosets → must be consistent.

⭐ 5. Quotient Groups (Related Topic)

If N is normal, then:

[
G/N = { gN : g \in G }
]

Operation defined by:

[
(gN)(hN) = (gh)N
]

This is well-defined only when N is normal.

⭐ 6. First Isomorphism Theorem (Key Result)

If ( f:G \to H ) is a homomorphism:

[
G/\ker(f) \cong \text{Im}(f)
]

Thus kernels are the building blocks of quotient groups.

⭐ 7. Applications of Normal Subgroups

✔ Building simpler groups via quotient groups
✔ Classifying groups
✔ Galois theory
✔ Solving polynomial equations
✔ Cryptography
✔ Symmetry analysis
✔ Geometry and physics (rotational symmetry groups)
✔ Automata theory and abstract machines

⭐ Exam-Oriented Summary

✔ Normal subgroup:

[
N \triangleleft G \iff gN = Ng \ \forall g \in G
]

✔ Equivalent conditions:

(gNg^{-1} = N)
Kernel of homomorphism
Closed under conjugation

✔ In abelian groups:

Every subgroup is normal.

✔ Use:

To form quotient groups (G/N).

✔ Important Results:

(A_n \triangleleft S_n)
If index = 2 → normal
First Isomorphism Theorem