Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by A_n or Alt(n).

For n > 1, the group A_n is the commutator subgroup of the symmetric group S_n with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : S_n → {1, −1} explained under symmetric group.

The group A_n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A₅ is the smallest non-abelian simple group, having order 60, and thus the smallest non-solvable group.

The group A₄ has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions { (), (12)(34), (13)(24), (14)(23) }, that is the kernel of the surjection of A₄ onto A₃ ≅ Z₃. We have the exact sequence V → A₄ → A₃ = Z₃. In Galois theory, this map, or rather the corresponding map S₄ → S₃, corresponds to associating the Lagrange resolvent cubic to a quartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.

As in the symmetric group, any two elements of A_n that are conjugate by an element of A_n must have the same cycle shape. The converse is not necessarily true, however. If the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape (Scott 1987, §11.1, p299).

Examples:

As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even permutations, alternating groups are subgroups of finite symmetric groups.

For n ≥ 3, A_n is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that A_n is simple for n ≥ 5.