Alternating group

In mathematics an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by A_n.

For instance: {1234, 1342, 1423, 2143, 2314, 2431, 3124, 3241, 3412, 4132, 4213, 4321} is the alternating group of degree 4.

For n > 1, the group A_n is a normal 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 iff n ≤ 3 and simple iff n = 3 or n ≥ 5. A₅ is the smallest non-abelian simple group.