Group Theory

Group axioms

A group (G, •) is a set G together with a binary operation • satisfying:

1. Closure — for all a, b ∈ G, a • b ∈ G. (The operation stays within G.)
2. Associativity — (a • b) • c = a • (b • c).
3. Identity — there exists e ∈ G such that e • a = a • e = a for all a.
4. Inverses — for each a ∈ G, there exists a⁻¹ ∈ G with a • a⁻¹ = a⁻¹ • a = e.

If additionally a • b = b • a for all a, b, the group is abelian (commutative).

Examples

Group	Operation	Identity	Order	Abelian?
(ℤ, +)	Addition	0	Infinite	Yes
(ℝ, +)	Addition	0	Infinite (continuum)	Yes
(ℤ/n, +)	Addition mod n	0	n	Yes (cyclic)
S₃	Composition of permutations	identity perm	6	No
D₄ (square symmetries)	Composition	do nothing	8	No
GL(n, ℝ)	Matrix multiplication	I (identity matrix)	Infinite	No (n ≥ 2)
SO(3) (3D rotations)	Composition	no rotation	Infinite (Lie group)	No
(ℚ\{0}, ×)	Multiplication	1	Infinite	Yes

Groups as symmetry

Every symmetric object has a symmetry group — the set of all transformations preserving it.

• Equilateral triangle. 3 rotations (0°, 120°, 240°) + 3 reflections = 6 elements. This is D₃ (dihedral group of order 6) ≅ S₃.
• Square. 4 rotations + 4 reflections = 8 elements. D₄ (dihedral group of order 8).
• Regular n-gon. n rotations + n reflections = 2n elements. Dihedral group Dₙ.
• Cube. 24 rotational symmetries (or 48 with reflections). The rotation group of the cube is isomorphic to S₄.
• Sphere. Rotation group SO(3), continuous (infinite-dimensional).

Lagrange's theorem

For a finite group G with subgroup H, the order of H divides the order of G:

|H| divides |G|

Consequences:

• A group of prime order p has no non-trivial subgroups — must be cyclic ℤ/p.
• The order of any element a divides |G| (since the cyclic subgroup generated by a is a subgroup).
• If |G| = 12, possible subgroup orders are 1, 2, 3, 4, 6, 12 — restricting structure significantly.

Lagrange's theorem is the foundation for further structure theorems (Cauchy's, Sylow's).

Galois theory — birth of group theory

Évariste Galois (1811-1832) invented group theory to answer a 300-year-old question — why is there no general formula for quintic equations?

Each polynomial p(x) has an associated Galois group — a permutation group acting on its roots.

• If the Galois group is solvable (a technical condition involving normal subgroups), the polynomial can be solved in radicals.
• For degree ≤ 4, Galois groups are subgroups of S₂, S₃, S₄ — all solvable. So formulas exist.
• For degree 5, generic Galois group is S₅, which contains the simple non-abelian group A₅. NOT solvable.
• Therefore — no general radical formula for quintic polynomials. Confirms Abel-Ruffini (1824).

This connection between abstract group theory and concrete polynomial solving is one of the deepest results in algebra.

Lie groups — continuous symmetries

A Lie group is a group that's also a smooth manifold (the operations are smooth/differentiable). Examples:

Lie group	Description	Dimension	Physical role
U(1)	Complex numbers e^(iθ)	1	Electromagnetism gauge symmetry
SU(2)	2×2 unitary matrices, det 1	3	Weak interaction; spin in QM
SU(3)	3×3 unitary, det 1	8	Strong interaction (color charge)
SO(3)	3D rotations	3	Rotational symmetry, angular momentum
SO(3,1)	Lorentz group	6	Special relativity

Lie groups underlie all of modern fundamental physics — the Standard Model is a U(1) × SU(2) × SU(3) gauge theory.

JavaScript — basic group operations

// Modular addition group ℤ/n
class CyclicGroup {
  constructor(n) { this.n = n; }
  add(a, b) { return (a + b) % this.n; }
  inverse(a) { return (this.n - a) % this.n; }
  identity() { return 0; }
  elements() { return Array.from({ length: this.n }, (_, i) => i); }
}

const Z6 = new CyclicGroup(6);
console.log(Z6.add(4, 5));      // 3 (since 9 mod 6 = 3)
console.log(Z6.inverse(2));     // 4 (since 2 + 4 = 6 ≡ 0)

// Permutation group: store permutations as arrays
function composePerms(p, q) {
  // (p ∘ q)(i) = p(q(i))
  return q.map(qi => p[qi]);
}

function inversePerm(p) {
  const inv = new Array(p.length);
  p.forEach((pi, i) => inv[pi] = i);
  return inv;
}

// All 6 permutations of {0, 1, 2} = S₃
const S3 = [
  [0, 1, 2], [0, 2, 1], [1, 0, 2],
  [1, 2, 0], [2, 0, 1], [2, 1, 0]
];

// Check non-abelian: (1 0 2) ∘ (2 1 0) vs (2 1 0) ∘ (1 0 2)
const a = [1, 0, 2];  // swap 0, 1
const b = [2, 1, 0];  // swap 0, 2
console.log(composePerms(a, b));  // [2, 0, 1]
console.log(composePerms(b, a));  // [1, 2, 0] — different!

// Check Lagrange: subgroup orders divide |S₃| = 6
const subgroupOrders = [1, 2, 3, 6];  // possible orders
console.log(subgroupOrders.every(d => 6 % d === 0));  // true

Where group theory shows up

• Pure algebra. Foundation for ring theory, field theory, Galois theory, representation theory. Modern algebra is unthinkable without group theory.
• Particle physics. Standard Model is gauge theory U(1) × SU(2) × SU(3). Particle classifications use group representations (Eightfold Way, quark model).
• Crystallography. Exactly 230 space groups classify all 3D crystal structures. Foundational for materials science, X-ray diffraction.
• Chemistry — molecular symmetry. Molecular vibrational modes and electronic states transform per the molecular symmetry group. Predicts spectroscopic selection rules.
• Cryptography. Elliptic-curve cryptography uses groups of points on elliptic curves. RSA uses multiplicative group of (ℤ/N)*. Discrete logarithm hardness is a group property.
• Computer science — Rubik's cube and puzzles. Solving algorithms use the group of cube transformations. The Rubik's cube group has order ~4.3 × 10¹⁹.
• Geometry — Klein's Erlangen program. Defines geometry by the group of allowed transformations. Euclidean geometry — Euclidean group; projective — projective group; hyperbolic — Lorentz group.

Common mistakes

• Forgetting closure. If the operation can produce elements outside G, it's not a group. (ℕ, +) is not a group — no inverses (no negative numbers in ℕ). (ℕ, −) — not closed (3 − 5 ∉ ℕ).
• Confusing left and right inverses. Group axioms require both — a • a⁻¹ = a⁻¹ • a = e. In group theory, left = right, but in semigroup theory or other structures, they may differ.
• Assuming all groups are abelian. Many natural groups (S₃, matrices, rotations in 3D) are non-abelian. Composition order matters.
• Treating identity as "the same" across groups. Each group has its own identity. The identity of (ℤ, +) is 0; of (ℚ\{0}, ×) is 1. Don't confuse them.
• Misunderstanding Lagrange. "Order of subgroup divides order of group." The CONVERSE is false in general — not every divisor d of |G| has a subgroup of order d (Sylow theorems give a partial converse for prime powers).
• Confusing isomorphism with equality. S₃ ≅ D₃ — same abstract structure, different presentations. Most problems care about isomorphism class, not specific representation.