Abelian vs Non-Abelian Groups

 CSIR NET & IIT JAM Quick Notes

1. Definitions 

Abelian Group

Group (G, ∗) where ab = ba for all a, b ∈ G

Non-Abelian Group

Group where ab ≠ ba for some a, b ∈ G

2. Quick Recognition Rules

ALWAYS Abelian:

✓ All cyclic groups (ℤ, ℤₙ) ✓ Groups of prime order p ✓ Groups of order p² (p prime) ✓ (ℤ, +), (ℚ, +), (ℝ, +), (ℂ, +) ✓ (ℚ*, ×), (ℝ*, ×), (ℂ*, ×) ✓ Klein 4-group V₄

ALWAYS Non-Abelian (n ≥ 3):

✗ Sₙ (Symmetric groups) ✗ Dₙ (Dihedral groups) ✗ GLₙ(ℝ) (Matrix groups) ✗ Quaternion group Q₈

3. Key Properties Comparison

PropertyAbelianNon-Abelian
ab = ba?YES (all)NO (some)
Z(G)= G⊂ G
All subgroups normal?YESNO
[G,G]{e}≠ {e}
Cayley tableSymmetricNot symmetric

4. Important Theorems (Exam Focus)

Theorem 1: Cyclic ⟹ Abelian

(Converse FALSE: V₄ is abelian but not cyclic)

Theorem 2: |G| = p (prime) ⟹ G cyclic ⟹ G abelian

Theorem 3: |G| = p² ⟹ G abelian

Theorem 4: If G, H abelian ⟹ G × H abelian

Theorem 5: G abelian, N ⊴ G ⟹ G/N abelian

Theorem 6: All subgroups of abelian groups are normal

5. Standard Examples (Memorize)

Abelian Examples:

1. ℤₙ = {0,1,2,...,n-1} under +

  • Cyclic: ⟨1⟩ = ℤₙ

2. Klein 4-group: V₄ = {e,a,b,c}

·  | e  a  b  c
---|------------
e  | e  a  b  c
a  | a  e  c  b
b  | b  c  e  a
c  | c  b  a  e
  • V₄ ≅ ℤ₂ × ℤ₂
  • All elements have order 2

3. ℤₘ × ℤₙ (always abelian)

Non-Abelian Examples:

1. S₃ = {e, (12), (13), (23), (123), (132)}

Quick proof:

  • (12)(13) = (132)
  • (13)(12) = (123)
  • (132) ≠ (123) ✓

2. D₃ (Triangle symmetries)

  • |D₃| = 6
  • D₃ ≅ S₃

3. Matrix Group:

A = [1 1]    B = [1 0]
    [0 1]        [1 1]

AB = [2 1]   BA = [1 1]
     [1 1]        [1 2]

AB ≠ BA ✓

4. Quaternions Q₈ = {±1, ±i, ±j, ±k}

  • ij = k, ji = -k
  • Non-abelian

6. Classification by Order

OrderNumber of GroupsAbelian?
p1YES (ℤₚ)
p²2YES (ℤₚ², ℤₚ×ℤₚ)
pq (q≢1 mod p)1YES (ℤₚᵧ)
pq (q≡1 mod p)21 abelian, 1 non-abelian
62ℤ₆ (abelian), S₃ (non-abelian)

7. Testing Methods (Exam Tricks)

Method 1: Cayley Table

  • Symmetric about diagonal → Abelian
  • Not symmetric → Non-abelian

Method 2: Find ONE counterexample

To prove non-abelian: find a, b where ab ≠ ba

Method 3: Use order

  • |G| = p → Abelian
  • |G| = p² → Abelian
  • |G| = 6 → Check if cyclic

Method 4: Check if cyclic

Cyclic → Abelian (always)

8. Center Z(G)

Definition: Z(G) = {g ∈ G : ga = ag for all a ∈ G}

Key Facts:

  • G abelian ⟺ Z(G) = G
  • G non-abelian ⟺ Z(G) ⊂ G (proper)
  • Z(G) is always normal in G
  • Z(G) is always abelian

Examples:

  • Z(S₃) = {e}
  • Z(D₃) = {e}
  • Z(Q₈) = {±1}

9. Commutator Subgroup [G,G]

Definition: [G,G] = ⟨aba⁻¹b⁻¹ : a,b ∈ G⟩

Key Facts:

  • G abelian ⟺ [G,G] = {e}
  • [G,G] ⊴ G (always normal)
  • G/[G,G] is always abelian

Examples:

  • [S₃,S₃] = A₃ = {e, (123), (132)}
  • [ℤₙ,ℤₙ] = {0}

10. Important Results for Exams

Result 1:

G/Z(G) cyclic ⟹ G abelian

Result 2:

|G| = pq, p < q primes:

  • If q ≢ 1 (mod p) → G abelian
  • If q ≡ 1 (mod p) → Two groups (1 abelian, 1 non-abelian)

Result 3:

p-groups have non-trivial center |G| = pⁿ ⟹ |Z(G)| ≥ p

Result 4:

All groups of order < 6 are abelian

Result 5:

Smallest non-abelian group: S₃ (order 6)

11. Common Exam Questions

Type 1: True/False

Q: Every group of order 15 is abelian. A: TRUE (15 = 3×5, and 5 ≢ 1 mod 3)

Q: Every abelian group is cyclic. A: FALSE (V₄ is abelian but not cyclic)

Q: S₄ is abelian. A: FALSE (Sₙ non-abelian for n ≥ 3)

Type 2: Prove Non-Abelian

Q: Show S₃ is non-abelian. A: (12)(13) = (132) ≠ (123) = (13)(12)

Type 3: Order-Based

Q: How many abelian groups of order 8? A: Three: ℤ₈, ℤ₄×ℤ₂, ℤ₂×ℤ₂×ℤ₂

Q: How many non-abelian groups of order 8? A: Two: D₄, Q₈

Type 4: Subgroups

Q: Can non-abelian group have abelian subgroup? A: YES (e.g., A₃ ⊂ S₃)

Q: In abelian group, all subgroups are? A: Normal

12. Quick Formulas

Center:

|G| = |Z(G)| · |G/Z(G)|

Commutator:

aba⁻¹b⁻¹ = e ⟺ ab = ba

Conjugacy:

G abelian ⟺ Each element in separate conjugacy class

13. Memory Tricks

Abelian Checklist:

  • Order p? → YES
  • Order p²? → YES
  • Cyclic? → YES
  • ℤₙ? → YES
  • Direct product of abelian? → YES

Non-Abelian Red Flags:

  • Sₙ (n≥3)? → YES
  • Dₙ (n≥3)? → YES
  • Matrices? → Usually YES
  • Order 6 and not cyclic? → YES (S₃)

14. Final Exam Tips

For CSIR NET:

  1. Know S₃ completely (all elements, products)
  2. Memorize: |G| = p, p² → abelian
  3. Practice center and commutator calculations
  4. Know Klein 4-group structure

For IIT JAM:

  1. Matrix group examples (GL₂)
  2. Dihedral groups Dₙ
  3. Quotient group properties
  4. Normal subgroups in abelian vs non-abelian

Most Asked:

  • S₃ properties and Cayley table
  • Prove group is/isn't abelian
  • Order-based classification
  • Center calculations
  • Subgroup normality

15. One-Liner Facts (Must Remember)

  1. Cyclic → Abelian (always)
  2. Abelian → Cyclic (NOT always)
  3. Order p → Cyclic → Abelian
  4. Order p² → Abelian
  5. Z(G) = G ⟺ G abelian
  6. [G,G] = {e} ⟺ G abelian
  7. All subgroups normal ⟺ G abelian (FALSE for infinite)
  8. Sₙ abelian only for n ≤ 2
  9. Smallest non-abelian group: order 6 (S₃)
  10. V₄: abelian, NOT cyclic
On my YouTube channel "Maths mastery with Dr. Upasana P Taneja," I regularly explore both types of groups and show when commutativity makes problems easier or when non-commutativity creates interesting complexity.

The universe of groups splits along this fundamental divide. Abelian groups are "well-behaved" and systematic. Non-abelian groups are wild and fascinating. Both are essential to modern mathematics!

Commute or not commute – that is the question!
Dr. Upasana Pahuja Taneja

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