Normal Subgroups: When Cosets Behave Nicely

 After teaching subgroups, students ask: "Are all subgroups created equal?" Not quite! Some subgroups are special – we call them normal subgroups, and they unlock one of group theory's most powerful concepts.

What is a Normal Subgroup?

A subgroup H of G is normal (written H ⊴ G) if:

gH = Hg for all g ∈ G

In other words: left cosets equal right cosets!

Alternative definition: gHg⁻¹ = H for all g ∈ G

This means H is closed under conjugation.

Why "Normal"?

Normal subgroups are "well-behaved" – they allow us to construct quotient groups, which are fundamental to understanding group structure.

Simple Examples

Example 1: Every Subgroup of an Abelian Group is Normal

In an abelian group, gH = Hg automatically (since gh = hg).

So in Z₆, every subgroup is normal:

  • {0}
  • {0, 3}
  • {0, 2, 4}
  • Z₆

Example 2: The Center is Always Normal

Z(G) = {z ∈ G | zg = gz for all g ∈ G}

Since center elements commute with everything, Z(G) ⊴ G always!

Example 3: Normal Subgroup in S₃

A₃ = {e, (123), (132)} is normal in S₃.

Check: For any σ ∈ S₃, σA₃σ⁻¹ = A₃

But H = {e, (12)} is NOT normal in S₃:

  • (123)H = {(123), (123)(12)} = {(123), (13)}
  • H(123) = {(123), (12)(123)} = {(123), (23)}

Since (13) ≠ (23), we have (123)H ≠ H(123) ✗

How to Check if H is Normal

Method 1: Verify gH = Hg for all g ∈ G

Method 2: Verify gHg⁻¹ = H for all g ∈ G

Method 3: Verify ghg⁻¹ ∈ H for all g ∈ G and h ∈ H

Method 4: If G is abelian, H is automatically normal

Method 5: If H has index 2, then H is normal

Important Properties

Property 1: The trivial subgroup {e} and G itself are always normal.

Property 2: If H is the only subgroup of its order, then H is normal.

Property 3: The kernel of any homomorphism is normal.

Property 4: If H ⊴ G and K ⊴ G, then H ∩ K ⊴ G.

Property 5: If [G:H] = 2 (index 2), then H ⊴ G.

Subgroups of Index 2

Theorem: Any subgroup of index 2 is automatically normal!

Example: In S₃, the subgroup A₃ has index [S₃:A₃] = 6/3 = 2, so A₃ ⊴ S₃.

Why? There are only two cosets: H and gH (for any g ∉ H). Similarly for right cosets. So the coset structures must match!

Non-Normal Subgroups

Example: Subgroups of S₃

S₃ has 6 subgroups, but only 3 are normal:

  • {e} ⊴ S₃ ✓
  • A₃ = {e, (123), (132)} ⊴ S₃ ✓
  • S₃ ⊴ S₃ ✓

NOT normal:

  • {e, (12)} ✗
  • {e, (13)} ✗
  • {e, (23)} ✗

Quotient Groups

When H ⊴ G, we can form the quotient group G/H:

Elements: The cosets {gH | g ∈ G}

Operation: (g₁H)(g₂H) = (g₁g₂)H

This only works when H is normal!

Example: S₃/A₃ has two elements: A₃ and (12)A₃

This quotient group is isomorphic to Z₂.

The Kernel Connection

Definition: For a homomorphism φ: G → G', the kernel is: ker(φ) = {g ∈ G | φ(g) = e'}

Theorem: The kernel of any homomorphism is a normal subgroup.

Example: Consider φ: ℤ → Z₆ defined by φ(n) = n (mod 6) ker(φ) = 6ℤ = {..., -6, 0, 6, 12, ...}

This is normal in ℤ (all subgroups of abelian groups are normal).

Practice Problem 1

Question: Is {e, (12)(34)} normal in S₄?

Answer: No

Explanation: This is not closed under conjugation. For example: (123){e, (12)(34)}(123)⁻¹ ≠ {e, (12)(34)}

Practice Problem 2

Question: How many normal subgroups does V₄ have?

A) 3 B) 5 C) 7 D) All subgroups

Answer: D) All subgroups

Explanation: V₄ is abelian, so all 5 subgroups are normal!

Simple Groups

Definition: A group G is simple if its only normal subgroups are {e} and G.

Simple groups are the "atoms" of group theory – they can't be broken down further!

Examples:

  • All groups of prime order are simple
  • A₅ (alternating group of order 60) is simple
  • All cyclic groups of prime order are simple

Multiple Choice Quiz

Question: Which statement is TRUE?

A) All subgroups are normal
B) All normal subgroups are abelian
C) All subgroups of abelian groups are normal
D) All normal subgroups have index 2

Answer: C) All subgroups of abelian groups are normal

Explanation:

  • A) False: {e, (12)} in S₃ is not normal
  • B) False: A₄ is a normal subgroup of S₄ but A₄ is non-abelian
  • C) True: In abelian groups, gH = Hg always
  • D) False: Many normal subgroups don't have index 2

Why Normal Subgroups Matter

In Group Structure: They allow us to build quotient groups and understand composition series.

In Homomorphisms: Kernels are always normal – connecting homomorphisms to quotient groups.

In Galois Theory: Normal subgroups correspond to normal field extensions.

In Classification: Understanding normal subgroups helps classify all groups.

The First Isomorphism Theorem

Theorem: If φ: G → G' is a homomorphism, then: G/ker(φ) ≅ Im(φ)

This beautiful result connects normal subgroups (kernels), quotient groups, and homomorphisms!

Key Takeaways

✓ Normal subgroups have matching left and right cosets
✓ They're closed under conjugation
✓ All subgroups of abelian groups are normal
✓ Subgroups of index 2 are always normal
✓ Kernels of homomorphisms are normal
✓ Normal subgroups enable quotient groups

Normal subgroups bridge the gap between subgroup structure and the larger theory of homomorphisms and quotient groups. They're not just "nice" subgroups – they're essential to understanding how groups decompose and relate to each other!

For more explorations of normal subgroups and quotient groups, check out my YouTube channel "Maths Mastery with Dr. Upasana P Taneja"!

When cosets align, beautiful structure emerges!
Dr. Upasana P Taneja

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