Alternating groups

The alternating group Aₙ is the group of even permutations of n points, of order n!/2 — the kernel of the sign homomorphism on the symmetric group Sₙ. It is simple for every n ≥ 5, the fact behind the unsolvability of the general quintic.

Definition

• Even permutations. A permutation is even if it is a product of an even number of transpositions (this parity is well defined). Aₙ is the set of even permutations of {1, …, n}.
• Kernel of the sign map. With sgn: Sₙ → {±1} ≅ C₂, Aₙ = ker(sgn). For n ≥ 2 the map is onto, so Aₙ is normal in Sₙ of index 2 and |Aₙ| = n!/2.
• Derived subgroup of Sₙ. Aₙ = [Sₙ, Sₙ] (see derived subgroup); the abelianization is Sₙ/Aₙ ≅ C₂.
• Generation. For n ≥ 3, Aₙ is generated by the 3-cycles.

Simplicity and solvability

• Aₙ is simple for every n ≥ 5. For n ≥ 5 it is therefore non-abelian and perfect (Aₙ = [Aₙ, Aₙ]).
• Aₙ (and Sₙ) is solvable exactly when n ≤ 4. The non-solvability of A₅ ≤ S₅ is the Galois-theoretic obstruction behind Abel–Ruffini: the general polynomial of degree ≥ 5 is not solvable by radicals.
• A₅, of order 60, is the smallest non-abelian simple group — every group of smaller order is either non-simple or cyclic of prime order.

Small cases

• A₄ (order 12). The Klein four-group V₄ = C₂ × C₂ is a proper nontrivial normal subgroup, so A₄ is not simple. A₄ has no subgroup of order 6 — the standard witness that the converse of Lagrange's theorem fails — and it is the smallest group that is not supersolvable.
• A₆ (order 360). Isomorphic to PSL(2,𝔽9) (see matrix groups). It is the unique exceptional member: its outer automorphism group is C₂ × C₂, and its Schur multiplier (see homology) is C₆ rather than the generic C₂.

Automorphisms

For n ≥ 4 the center Z(Aₙ) is trivial, so Inn(Aₙ) ≅ Aₙ.

• Aut(Aₙ) ≅ Sₙ for n ≥ 4, with the single exception n = 6. Away from n = 6 this gives Out(Aₙ) ≅ Sₙ/Aₙ ≅ C₂.
• n = 6: |Aut(A₆)| = 1440 and Out(A₆) ≅ C₂ × C₂. A₆ is the only alternating group with an outer automorphism beyond the one induced by Sₙ.
• Schur multiplier. M(Aₙ) ≅ C₂ for n ≥ 5, exceptionally C₆ for n = 6 and n = 7.

Examples

• A₃ ≅ C₃ — order 3, the smallest nontrivial case.
• A₄ — order 12; the rotation group of the regular tetrahedron.
• A₅ — order 60; simple; the rotation group of the icosahedron/dodecahedron; ≅ PSL(2,𝔽4) ≅ PSL(2,𝔽5).
• A₆ — order 360; simple; ≅ PSL(2,𝔽9).

See also

• Symmetric group — Aₙ is its index-2 subgroup and derived subgroup.
• Simple group, perfect group — both hold for Aₙ, n ≥ 5.
• Solvable group — Aₙ solvable iff n ≤ 4.
• Matrix groups — the isomorphisms A₅ ≅ PSL(2,𝔽4) ≅ PSL(2,𝔽5) and A₆ ≅ PSL(2,𝔽9).
• Lagrange's theorem — A₄ has no subgroup of order 6.
• Wikipedia: Alternating group