Alternating group

The alternating group Aₙ consists of the even permutations of an n-element set, with composition as its operation. For n ≥ 2, it is the kernel of the sign homomorphism from the symmetric group Sₙ to the two-element group.

Permutation group: A group whose elements are permutations of a set and whose operation is composition. Aₙ is a permutation group consisting of exactly the even permutations.
Sign homomorphism: The homomorphism from Sₙ to {+1, −1} that assigns each permutation its parity sign. Its kernel is precisely the alternating group.
A₃: The alternating group on three elements, a cyclic group of order three. It is the smallest nontrivial alternating group and illustrates generation by a 3-cycle.
Group simplicity: The property of having no normal subgroups other than the identity subgroup and the whole group. Aₙ is simple for n ≥ 5, making these groups central examples in finite group theory.
Symmetric group: The group Sₙ of all permutations of an n-element set, under composition. Aₙ is the index-two subgroup of Sₙ formed by even permutations.
Cycle notation: A notation that writes a permutation as a product of disjoint cycles. Cycle decompositions make membership in Aₙ easy to test.
A₄: The alternating group on four elements, of order twelve. Its normal Klein four subgroup makes it structurally different from A₅ and larger groups.
Finite simple group: A finite group with no proper nontrivial normal subgroups. The groups Aₙ for n ≥ 5 form an infinite family of finite simple groups.
Even permutation: A permutation expressible as a product of an even number of transpositions. Evenness is the defining membership condition for Aₙ.
3-cycle: A permutation that cyclically moves three elements and fixes all others. Every 3-cycle is even, and 3-cycles generate Aₙ.

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