# Symmetric Group

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

## Summary
Sₙ = all n! rearrangements of n objects. Every finite group embeds in some Sₙ (Cayley's theorem). S₅'s non-solvability is why no quintic formula exists.

## Cycle notation and transpositions
Every permutation decomposes into disjoint cycles. Every permutation is a product of transpositions (2-cycles); parity (even/odd) is always well-defined.

## Alternating group Aₙ
Even permutations form Aₙ, index 2 in Sₙ. Aₙ is simple for n≥5 — the crucial fact.

## Cayley's theorem (1854)
Every group of order n embeds as a subgroup of Sₙ.

## Abel-Ruffini theorem
Sₙ is solvable for n≤4 (hence quadratic/cubic/quartic formulas exist) but NOT solvable for n≥5 (since Aₙ is simple, non-abelian) — proven by Ruffini (1799, gap), completed by Abel (1824), explained conceptually by Galois.

## Representation theory
Irreducible representations of Sₙ correspond bijectively to partitions of n (Young tableaux, Alfred Young 1900).

## Physics connection
Symmetric group structure underlies the Pauli exclusion principle for fermions.

## Sources
### Tier 1
- Dummit, D.S. and Foote, R.M. (2004). *Abstract Algebra*. 3rd ed.
- Fulton, W. and Harris, J. (1991). *Representation Theory*.
### Tier 2
- Rotman, J.J. (1995). *An Introduction to the Theory of Groups*. 4th ed.

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