An identity permutation is an even permutation as (1)=(1 2)(1 2).

The composition of two even permutations is again an even permutation, and so is the inverse of an even permutation: the even permutations of *n* letters form a group, the alternating group on *n* letters, denoted by A_{n}. This is a subgroup of the symmetric group S_{n} and contains *n*/2 permutations.

An **odd permutation** is a permutation which is not an even permutation, equivalently, it is a product by odd number of transpositions.

See fifteen puzzle for a classic application.