Editing Parity of a permutation (section)

== Properties ==
The identity permutation is an even permutation.<ref name="Jacobson" /> An even permutation can be obtained as the composition of an [[even and odd numbers|even number]] (and only an even number) of exchanges (called [[transposition (mathematics)|transposition]]s) of two elements, while an odd permutation can be obtained by (only) an odd number of transpositions.

The following rules follow directly from the corresponding rules about addition of integers:<ref name="Jacobson" />
* 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<sub>''n''</sub> of all permutations of the set {1, ..., ''n''}, we can conclude that the map
:{{math|1=sgn: S<sub>''n''</sub> → {−1, 1} }}<!-- do not use <math> here.  see https://bugzilla.wikimedia.org/show_bug.cgi?id=1594#c4 -->
that assigns to every permutation its signature is a [[group homomorphism]].<ref>Rotman (1995), [{{Google books|plainurl=y|id=lYrsiaHSHKcC|page=9|text=sgn}} p. 9, Theorem 1.6.]</ref>

Furthermore, we see that the even permutations form a [[subgroup]] of S<sub>''n''</sub>.<ref name="Jacobson" /> This is the [[alternating group]] on ''n'' letters, denoted by A<sub>''n''</sub>.<ref name="Jacobson_a">Jacobson (2009), p. 51.</ref> It is the [[Kernel (algebra)|kernel]] of the homomorphism sgn.<ref>Goodman, [{{Google books|plainurl=y|id=l1TKk4InOQ4C|page=116|text=kernel of the sign homomorphism}} p. 116, definition 2.4.21]</ref> The odd permutations cannot form a subgroup, since the composite of two odd permutations is even, but they form a [[coset]] of A<sub>''n''</sub> (in S<sub>''n''</sub>).<ref>Meijer & Bauer (2004), [{{Google books|plainurl=y|id=ZakN8Y7dcC8C|page=72|text=these permutations do not form a subgroup since the product of two odd permutations is even}} p. 72]</ref>

If {{nowrap|''n'' > 1}}, then there are just as many even permutations in S<sub>''n''</sub> as there are odd ones;<ref name="Jacobson_a" /> consequently, A<sub>''n''</sub> contains [[factorial|''n''!]]/2 permutations. (The reason is that if ''σ'' is even then {{nowrap|(1  2)''σ''}} is odd, and if ''σ'' is odd then {{nowrap|(1  2)''σ''}} is even, and these two maps are inverse to each other.)<ref name="Jacobson_a" />

A [[cyclic permutation|cycle]] is even if and only if its length is odd. This follows from formulas like
:<math>(a\ b\ c\ d\ e)=(d\ e)(c\ e)(b\ e)(a\ e)\text{ or }(a\ b)(b\ c)(c\ d)(d\ e).</math>
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 the same as the parity of the permutation.

Every permutation of odd [[order (group theory)|order]] must be even. The permutation {{nowrap|(1 2)(3 4)}} in A<sub>4</sub> shows that the converse is not true in general.