The Symmetric Group — Mathematics Knowledge System

Overview

The symmetric group Sₙ is the group of all bijections (permutations) of a set of n elements under composition, with order n!. Every finite group embeds into some symmetric group (Cayley's theorem), making Sₙ in a precise sense the universal finite group. Sₙ is generated by transpositions, splits into even and odd permutations via the sign homomorphism, and its subgroup of even permutations, the alternating group Aₙ, is simple for n ≥ 5 — the fact underlying Galois's proof that the general quintic has no solution by radicals.

Intuition

Sₙ contains every conceivable way to shuffle n labeled objects, so studying it is studying 'all possible rearrangements' in the most general sense — which is why Cayley's theorem can embed ANY finite group into some Sₙ (let G act on itself by left multiplication, and each element becomes a permutation of G's own elements). The sign function is a robust parity check: no matter how you write a permutation as a product of swaps, the number of swaps used is always even or always odd for that permutation, splitting Sₙ cleanly into two equal halves.

Formal Definition

Definition

Sₙ is the set of all bijections σ: {1,...,n} → {1,...,n} under function composition:

|S_n| = n!

Order of the symmetric group

\operatorname{sgn}: S_n \to \{\pm 1\}, \qquad \operatorname{sgn}(\sigma) = (-1)^{\#\text{transpositions}}

Sign homomorphism (well-defined regardless of decomposition chosen)

A_n = \ker(\operatorname{sgn}) \trianglelefteq S_n, \quad [S_n : A_n] = 2

Alternating group as kernel of sgn, index 2

\text{Every permutation is a product of disjoint cycles.}

Cycle decomposition theorem

Worked Examples

A 3-cycle decomposes into 2 transpositions: (1 3 2) = (1 2)(1 3).
$(1\,3\,2) = (1\,2)(1\,3)$
The 2-cycle (4 5) is already a single transposition.
$(4\,5) = (4\,5)$
Total: 2 + 1 = 3 transpositions, an odd number, so sgn(σ) = -1.
$\operatorname{sgn}(\sigma) = (-1)^3 = -1$

Answer: σ = (1 2)(1 3)(4 5), using 3 transpositions, so σ is odd with sgn(σ) = −1.

Practice Problems

Difficulty 4/10

Is the 5-cycle (1 2 3 4 5) an even or odd permutation?

Difficulty 5/10

State Cayley's theorem and explain briefly why it implies every group of order n embeds in S_n.

Difficulty 6/10

Explain why A₅ being simple (having no nontrivial normal subgroups) is significant for the solvability of the quintic by radicals.

Quiz

Summary

Sₙ, the group of all permutations of n objects under composition, has order n! and every finite group embeds into some Sₙ (Cayley's theorem).
The sign homomorphism sgn: Sₙ → {±1} splits Sₙ into even and odd permutations; Aₙ = ker(sgn) has index 2.
Aₙ is simple for n ≥ 5, the group-theoretic fact behind Abel–Ruffini: the general quintic has no solution by radicals.

References

WebsiteWikipedia — Symmetric group