What Is a Group?
In mathematics, a group is one of the most fundamental algebraic structures—a set equipped with an operation that combines elements in a way that satisfies four specific properties.
But that dry definition misses the poetry. Groups are the mathematical language of symmetry. Anywhere you find symmetry—in crystals, in physics, in music, in the solutions of equations—you find groups.
Group theory is the study of symmetry. When we are dealing with an object that appears symmetric, group theory can help us analyze it.
Mark Armstrong, Groups and Symmetry
The concept emerged in the early 19th century from two seemingly unrelated problems: Évariste Galois' work on polynomial equations and the study of geometric transformations. Galois, who died in a duel at age 20, left behind ideas so profound that they took decades to fully understand.
A group captures the essence of "things you can do that can be undone, combined, and have a do-nothing option." Rotate a square, flip it, rotate it back—these operations form a group. Add numbers, subtract them—that's a group too. The abstraction reveals deep connections between wildly different mathematical objects.
The Four Axioms
A group (G, ∗) consists of a set G and a binary operation ∗ satisfying:
1. CLOSURE
For all a, b ∈ G: a ∗ b ∈ G
"Combining two elements gives another element in the set"
2. ASSOCIATIVITY
For all a, b, c ∈ G: (a ∗ b) ∗ c = a ∗ (b ∗ c)
"Grouping doesn't matter"
3. IDENTITY
There exists e ∈ G such that for all a ∈ G: e ∗ a = a ∗ e = a
"There's a do-nothing element"
4. INVERSE
For every a ∈ G, there exists a⁻¹ ∈ G such that: a ∗ a⁻¹ = a⁻¹ ∗ a = e
"Every action can be undone"
// Note: Commutativity (a ∗ b = b ∗ a) is NOT required
// Groups with commutativity are called "abelian" groupsWhat's NOT Required: Commutativity
Many groups are non-abelian—order matters. Consider rotating a book: rotating 90° around the x-axis, then 90° around the y-axis, gives a different result than doing those rotations in the opposite order.
Examples: From Integers to Rubik's Cubes
The Integers Under Addition: (ℤ, +)
The most familiar group:
Set: ℤ = {..., -2, -1, 0, 1, 2, 3, ...}
Operation: Addition (+)
Closure: Any integer + any integer = integer ✓
Associativity: (a + b) + c = a + (b + c) ✓
Identity: 0 (since a + 0 = 0 + a = a) ✓
Inverse: For any a, inverse is -a (since a + (-a) = 0) ✓
This group is abelian (commutative): a + b = b + aThe Symmetric Group S_n: Permutations
The set of all permutations of n objects forms a group under composition. This is fundamental to understanding symmetry.
Elements of S₃ (6 total = 3! = 3×2×1):
e = (1)(2)(3) [identity: leave everything alone]
σ₁ = (1 2 3) [cycle: 1→2→3→1]
σ₂ = (1 3 2) [cycle: 1→3→2→1]
τ₁ = (1 2) [swap 1 and 2, fix 3]
τ₂ = (1 3) [swap 1 and 3, fix 2]
τ₃ = (2 3) [swap 2 and 3, fix 1]
Composition Example:
τ₁ ∘ σ₁ means: first apply σ₁, then apply τ₁
σ₁: 1→2, 2→3, 3→1
τ₁: 1→2, 2→1, 3→3
σ₁ then τ₁:
1 →(σ₁)→ 2 →(τ₁)→ 1
2 →(σ₁)→ 3 →(τ₁)→ 3
3 →(σ₁)→ 1 →(τ₁)→ 2
Result: (2 3) = τ₃
S₃ is non-abelian: τ₁ ∘ σ₁ ≠ σ₁ ∘ τ₁The Rubik's Cube Group
The Rubik's Cube is a physical manifestation of a group—specifically, a subgroup of the symmetric group on 48 "facelets" (not counting centers).
The Rubik's Cube Group:
Generators: {F, B, L, R, U, D} (six face rotations)
Order: 43,252,003,274,489,856,000 ≈ 4.3 × 10¹⁹
That's 43 quintillion possible positions!
Structure:
The group is a semidirect product:
G ≅ (C₃⁷ × C₂¹¹) ⋊ (S₈ × S₁₂)
God's Number:
Maximum moves needed from any position to solved: 20 (half-turn metric)
Proven by computer in 2010 (35 CPU-years of computation)Interactive Group Explorer
LIVEExplore different groups and their Cayley tables. Click cells to see how the group operation works.
| ∘ | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
Subgroups and Cosets
Subgroups
A subgroup H of G is a subset that is itself a group under the same operation.
One-Step Subgroup Test:
H ≤ G if and only if:
1. H ≠ ∅ (H is non-empty)
2. For all a, b ∈ H: a ∗ b⁻¹ ∈ H
Example: Even integers in (ℤ, +)
Let H = 2ℤ = {..., -4, -2, 0, 2, 4, ...}
Check: For a = 2m, b = 2n (even integers):
a + (-b) = 2m + (-2n) = 2(m - n) ∈ 2ℤ ✓
So 2ℤ ≤ ℤ (even integers form a subgroup)Cosets and Lagrange's Theorem
If H is a subgroup of G and g ∈ G, the left coset of H containing g is:
If H is a subgroup of a finite group G, then |H| divides |G|.
In notation: |G| = |H| × [G : H]
Where [G : H] is the index (number of distinct cosets).
Homomorphisms: Structure-Preserving Maps
A homomorphism is a function between groups that respects the group structure:
Monomorphism: Injective (one-to-one) homomorphism
Epimorphism: Surjective (onto) homomorphism
Isomorphism: Bijective homomorphism (groups are "the same")
Automorphism: Isomorphism from G to itself
Example: exp as a homomorphism
φ: (ℝ, +) → (ℝ⁺, ×)
φ(x) = eˣ
Check: φ(a + b) = e^(a+b) = eᵃ × eᵇ = φ(a) × φ(b) ✓
This is an isomorphism!
The additive reals are "the same" as positive multiplicative reals.Groups as Symmetry
This is where group theory becomes beautiful. Every symmetry of an object corresponds to a group element. The group structure captures how symmetries combine.
Symmetries of a Square: The Dihedral Group D₄
Elements (8 total):
Rotations:
e = identity (0°)
r = rotation by 90° counterclockwise
r² = rotation by 180°
r³ = rotation by 270°
Reflections:
s₁ = reflection across horizontal axis
s₂ = reflection across vertical axis
d₁ = reflection across main diagonal
d₂ = reflection across anti-diagonal
Group Structure:
r⁴ = e (four rotations return to start)
s² = e (reflecting twice = identity)
srs = r⁻¹ = r³ (fundamental relation)
Non-abelian: r ∘ s₁ = d₁, but s₁ ∘ r = d₂Applications: Physics to Cryptography
Physics: Conservation Laws
Emmy Noether's theorem (1915) established a profound connection: every continuous symmetry corresponds to a conservation law.
Symmetry Conservation Law
────────────────────────────────────────────
Time translation Energy
Space translation Momentum
Rotation Angular momentum
Phase (U(1) gauge) Electric charge
SU(3) color gauge Color charge (QCD)
// The Standard Model of particle physics is built on
// the gauge group SU(3) × SU(2) × U(1)
// Group theory IS the language of fundamental physicsChemistry: Molecular Symmetry
Molecule Point Group Symmetry Elements
─────────────────────────────────────────────
H₂O C₂ᵥ E, C₂, σᵥ, σᵥ'
NH₃ C₃ᵥ E, 2C₃, 3σᵥ
CH₄ Tᵈ E, 8C₃, 3C₂, 6S₄, 6σᵈ
SF₆ Oₕ 48 elements
C₆₀ (Bucky) Iₕ 120 elements (icosahedral)
// Point group determines:
// - IR and Raman active modes
// - Allowed electronic transitions
// - Molecular orbitalsCryptography: Elliptic Curve Groups
Modern cryptography relies heavily on group theory. Elliptic curve cryptography uses the group of points on an elliptic curve:
Curve: y² = x³ + ax + b (over a finite field F_p)
Group Operation:
Points on the curve form a group under "addition"
P + Q is defined geometrically (chord-and-tangent)
Identity: Point at infinity O
Security:
Given P and Q = nP (where n is secret)
Finding n is the "Elliptic Curve Discrete Log Problem"
Believed to be hard (no efficient algorithm known)
Applications:
- ECDSA (Bitcoin, Ethereum signatures)
- ECDH (Key exchange)
- TLS 1.3 (most HTTPS connections)
// 256-bit elliptic curve ≈ 3072-bit RSA security
// Much smaller keys = faster, less bandwidthConclusion: The Universal Language
Group theory reveals that the same mathematical structure underlies wildly different phenomena:
- The rotations of a molecule and the symmetries of a quantum field
- The shuffles of a deck of cards and the transformations of space-time
- The operations on a Rubik's Cube and the structure of error-correcting codes
Group theory succeeds because it captures something fundamental about the universe: symmetry is not just aesthetic—it's structural. The laws of physics are what they are because of the symmetries nature respects. Understanding groups is understanding the skeleton upon which physical reality hangs.
Welcome to the group. The operation is learning. The identity is curiosity. And there's always an inverse—a new question for every answer.