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**, or the

*n***alternating group on**and denoted by A

*n*letters_{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.
A_{5} is the smallest non-abelian simple group.