Symmetric groups

The symmetric group Sₙ is the group of all permutations (bijections) of an n-element set, under composition, of order n!. Every finite group of order n embeds in Sₙ — Cayley's theorem — so the Sₙ are the universal hosts of finite group theory.

Definition

For n ≥ 1, fix the set {1, 2, …, n}.

Sₙ = { bijections σ : {1,…,n} → {1,…,n} },   product = composition,   |Sₙ| = n!.

Cayley's theorem. The left-regular action g ↦ (x ↦ gx) embeds any group G with |G| = n as a subgroup of Sₙ.

Generators

Sₙ is generated by its transpositions (2-cycles), and more economically by:

• the n − 1 adjacent transpositions (i, i+1) — the Coxeter generators making Sₙ the Coxeter group of type Aₙ₋₁;
• a single transposition together with a full cycle: ⟨ (1 2), (1 2 … n) ⟩ = Sₙ — just two generators.

Structure

• Sign homomorphism. For n ≥ 2, sgn : Sₙ → C₂ sends even permutations to +1, odd to −1. Its kernel is the alternating group Aₙ, normal of index 2 with |Aₙ| = n!/2. The map splits (any transposition is a complement), so Sₙ = Aₙ ⋊ C₂ — a semidirect product.
• Derived subgroup. [Sₙ, Sₙ] = Aₙ for n ≥ 2, hence the abelianization is Sₙ/Aₙ ≅ C₂.
• Center. Z(Sₙ) = {e} for n ≥ 3 (trivial). Z(S₂) = S₂.
• Conjugacy classes. Two permutations are conjugate iff they share a cycle type, so the classes correspond to the partitions of n; their number is p(n), the partition function: p(1..5) = 1, 2, 3, 5, 7. The complex irreducible representations are also indexed by partitions of n, so there are exactly p(n) of them.
• Rational group. Every character value of Sₙ is an integer, so its character table is integer-valued.
• Automorphisms. Sₙ is a complete group — trivial center and no outer automorphisms, Aut(Sₙ) = Inn(Sₙ) ≅ Sₙ — for every n ≥ 3 except n = 6. S₆ carries an exceptional outer automorphism with |Out(S₆)| = 2, and S₆ is the only symmetric group with a nontrivial outer automorphism.
• Solvability. Sₙ is solvable iff n ≤ 4. For n ≥ 5, Sₙ ⊇ Aₙ, which is simple and non-abelian, so the series stops: Sₙ ⊵ Aₙ ⊵ {e} with the non-abelian simple factor Aₙ.

Examples

• S₃ ≅ D₃ — the dihedral symmetries of the triangle; order 6, the smallest non-abelian group.
• S₄ — order 24, isomorphic to the rotation group of the cube (acting on its 4 space diagonals); the largest solvable Sₙ.
• S₅ — order 120; the first nonsolvable symmetric group, as its derived subgroup A₅ is simple.

See also

• Alternating group — the index-2 subgroup Aₙ = ker(sgn).
• Cayley's theorem — every finite group embeds in some Sₙ.
• Conjugacy — classes of Sₙ are the cycle types (partitions).
• Rational group — the character table of Sₙ is integer-valued.
• Automorphism — Sₙ is complete except for S₆.
• Solvable group — Sₙ is solvable iff n ≤ 4.
• Wikipedia: Symmetric group