# Group

**Type:** Concept  
**Domain:** Algebra  
**Codex URL:** /mathematics/algebra/group/  
**Entry status:** Live — v1.0 (2026-05-27)

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## Curious Level

A group is a set of things combined with an operation that obeys four rules: closure (combining two elements gives another in the set), associativity (bracketing order doesn't change the result), identity (there is a "do nothing" element), and inverses (every element can be undone).

The integers under addition form a group. The rotations of an equilateral triangle form a group. The symmetries of any shape form a group.

The term was introduced by Évariste Galois around 1830, in work he wrote the night before dying in a duel at age 20. He used groups to prove there is no formula for the roots of a general degree-5 polynomial.

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## Exploring Level

### Definition

A **group** is an ordered pair (G, ★) where ★ is a binary operation on G satisfying:

1. **Associativity:** (a★b)★c = a★(b★c) for all a, b, c ∈ G
2. **Identity:** ∃e ∈ G such that e★a = a★e = a for all a ∈ G
3. **Inverses:** for each a ∈ G, ∃a⁻¹ ∈ G such that a★a⁻¹ = a⁻¹★a = e

If a★b = b★a for all a, b, the group is **abelian**.

### Examples

| Group | Set | Operation | Identity | Abelian? |
|-------|-----|-----------|----------|----------|
| ℤ | Integers | + | 0 | Yes |
| ℚ* | Nonzero rationals | × | 1 | Yes |
| ℤ/nℤ | Integers mod n | + mod n | 0 | Yes |
| Sₙ | Permutations of n elements | Composition | Identity perm. | n ≤ 2 only |
| Dₙ | Symmetries of regular n-gon | Composition | Identity | n ≤ 2 only |
| GL(n, ℝ) | Invertible n×n real matrices | Matrix mult. | Identity matrix | n ≥ 2: No |

### Lagrange's Theorem

If G is a finite group and H ≤ G is a subgroup, then |H| divides |G|.

### First Isomorphism Theorem

If φ : G → H is a homomorphism, then G / ker(φ) ≅ im(φ).

### Galois Theory

Galois associated to each polynomial a group (Galois group) that encodes the symmetries of its roots. A polynomial is solvable by radicals if and only if its Galois group is solvable. Since S₅ is not solvable, there is no radical formula for the general quintic.

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## Deep Dive

### Historical development

Lagrange (1771) observed permutation constraints on polynomial roots. Abel (1824) proved quintic insolubility. Galois (1830–1832) introduced the term "group" and the decisive theory. The abstract axiomatic definition emerged in the 1880s–1890s (Cayley, Dyck, Weber).

### Classification of Finite Simple Groups

Every finite simple group belongs to one of: cyclic groups ℤ/pℤ, alternating groups Aₙ (n ≥ 5), 16 infinite families of groups of Lie type, or 26 sporadic groups. Proof completed 2004 — spanning tens of thousands of pages. The largest sporadic group (Monster) has order ~8 × 10⁵³; connected to modular forms and string theory (Monstrous Moonshine, proved by Borcherds 1992).

### Groups in physics

Noether's theorem (1915): every continuous symmetry corresponds to a conserved quantity. The Standard Model gauge group is U(1) × SU(2) × SU(3). Representation theory of groups is the language of quantum mechanics.

### Free groups and undecidability

Every group is a quotient of a free group. The word problem — deciding if two words in generators represent the same element — is undecidable in general (Novikov, 1955).

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## Sources

### Tier 1
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed. John Wiley & Sons. [Chapters 1–6]
- Lang, S. (2002). *Algebra*. Revised 3rd ed. Springer.

### Tier 2
- Rotman, J.J. (1995). *An Introduction to the Theory of Groups*. 4th ed. Springer.

### Tier 3
- Stewart, I. (2015). *Galois Theory*. 4th ed. CRC Press.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*