Parity of A Permutation - Properties

Properties

The identity permutation is an even permutation. An even permutation can be obtained from the identity permutation by an even number of exchanges (called transpositions) of two elements, while an odd permutation can be obtained by an odd number of transpositions.

The following rules follow directly from the corresponding rules about addition of integers:

• the composition of two even permutations is even
• the composition of two odd permutations is even
• the composition of an odd and an even permutation is odd

From these it follows that

• the inverse of every even permutation is even
• the inverse of every odd permutation is odd

Considering the symmetric group S_n of all permutations of the set {1, ..., n}, we can conclude that the map

sgn: S_n → {−1, 1} 

that assigns to every permutation its signature is a group homomorphism.

Furthermore, we see that the even permutations form a subgroup of S_n. This is the alternating group on n letters, denoted by A_n. It is the kernel of the homomorphism sgn. The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a coset of A_n (in S_n).

If n > 1 , then there are just as many even permutations in S_n as there are odd ones; consequently, A_n contains n!/2 permutations.

A cycle is even if and only if its length is odd. This follows from formulas like

(a b c d e) = (d e) (c e) (b e) (a e)

In practice, in order to determine whether a given permutation is even or odd, one writes the permutation as a product of disjoint cycles. The permutation is odd if and only if this factorization contains an odd number of even-length cycles.

Another method for determining whether a given permutation is even or odd is to construct the corresponding Permutation matrix and compute its determinant. The value of the determinant is same as the parity of the permutation.

Every permutation of odd order must be even. The permutation (12)(34) in A₄ shows that the converse is not true in general.