Parity of a permutation

In mathematics, when X is a finite set of at least two elements, the permutations of X (i.e. the bijective mappings from X to X) fall into two classes of equal size: the even permutations and the odd permutations. If any total ordering of X is fixed, the parity (oddness or evenness) of a permutation σ of X can be defined as the parity of the number of inversions for σ, i.e., of pairs of elements x,y of X such that x < y and σ(x) > σ(y).
The sign or signature of a permutation σ is denoted sgn(σ) and defined as +1 if σ is even and −1 if σ is odd. The signature defines the alternating character of the symmetric group S_{n}. Another notation for the sign of a permutation is given by the more general LeviCivita symbol (), which is defined for all maps from X to X, and has value zero for nonbijective maps.
The sign of a permutation can be explicitly expressed as
 sgn(σ) = ( − 1)^{N(σ)}
where N(σ) is the number of inversions in σ.
Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as
 sgn(σ) = ( − 1)^{m}
where m is the number of transpositions in the decomposition. Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is welldefined.^{[1]}
Contents
Example
Consider the permutation σ of the set {1,2,3,4,5} which turns the initial arrangement 12345 into 34521. It can be obtained by three transpositions: first exchange the places of 1 and 3, then exchange the places of 2 and 4, and finally exchange the places of 1 and 5. This shows that the given permutation σ is odd. Using the notation explained in the permutation article, we can write
There are many other ways of writing σ as a composition of transpositions, for instance
but it is impossible to write it as a product of an even number of transpositions.
Properties
The identity permutation is an even permutation.^{[1]} 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:^{[1]}
 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
that assigns to every permutation its signature is a group homomorphism.^{[2]}
Furthermore, we see that the even permutations form a subgroup of S_{n}.^{[1]} This is the alternating group on n letters, denoted by A_{n}.^{[3]} It is the kernel of the homomorphism sgn.^{[4]} 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}).^{[5]}
If n>1, then there are just as many even permutations in S_{n} as there are odd ones;^{[3]} consequently, A_{n} contains n!/2 permutations. [The reason: if σ is even, then (12)σ is odd; if σ is odd, then (12)σ is even; the two maps are inverse to each other.]^{[3]}
A cycle is even if and only if its length is odd. This follows from formulas like
 (a b c d e) = (a e) (b e) (c e) (d 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 evenlength 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 converse is not true in general.
Equivalence of the two definitions
Proof 1
Every permutation can be produced by a sequence of transpositions (2element exchanges): with the first transposition we put the first element of the permutation in its proper place, the second transposition puts the second element right etc. Given a permutation σ, we can write it as a product of transpositions in many different ways. We want to show that either all of those decompositions have an even number of transpositions, or all have an odd number.
Suppose we have two such decompositions:
 σ = T_{1} T_{2} ... T_{k}
 σ = Q_{1} Q_{2} ... Q_{m}.
We want to show that k and m are either both even, or both odd.
Every transposition can be written as a product of an odd number of transpositions of adjacent elements, e.g.
 (2 5) = (2 3)(3 4)(4 5)(4 3)(3 2)
If we decompose in this way each of the transpositions T_{1}...T_{k} and Q_{1}...Q_{m} above into an odd number of adjacent transpositions, we get the new decompositions:
 σ = T_{1'} T_{2'} ... T_{k'}
 σ = Q_{1'} Q_{2'} ... Q_{m'}
where all of the T_{1'}...T_{k'} Q_{1'}...Q_{m'} are adjacent, k − k' is even, and m − m' is even.
Now compose the inverse of T_{1} with σ. T_{1} is the transposition (i, i + 1) of two adjacent numbers, so, compared to σ, the new permutation σ(i, i + 1) will have exactly one inversion pair less (in case (i,i + 1) was an inversion pair for σ) or more (in case (i, i + 1) was not an inversion pair). Then apply the inverses of T_{2}, T_{3}, ... T_{k} in the same way, "unraveling" the permutation σ. At the end we get the identity permutation, whose N is zero. This means that the original N(σ) less k is even.
We can do the same thing with the other decomposition, Q1...Qm, and it will turn out that the original N(σ) less m is even.
Therefore, m − k is even, as we wanted to show.
We can now define the permutation σ to be even if N(σ) is an even number, and odd if N(σ) is odd. This coincides with the definition given earlier but it is now clear that every permutation is either even or odd.
Proof 2
An alternative proof uses the polynomial
So for instance in the case n = 3, we have
Now for a given permutation σ of the numbers {1,...,n}, we define
Since the polynomial has the same factors as except for their signs, if follows that sgn(σ) is either +1 or −1. Furthermore, if σ and τ are two permutations, we see that
Since with this definition it is furthermore clear that any transposition of two adjacent elements has signature −1, we do indeed recover the signature as defined earlier.
Proof 3
A third approach uses the presentation of the group S_{n} in terms of generators and relations
 for all i
 for all i < n − 1
 if i − j ≥ 2.
[Here the generator τ_{i} represents the transposition (i, i + 1).] All relations keep the length of a word the same or change it by two. Starting with an evenlength word will thus always result in an evenlength word after using the relations, and similarly for oddlength words. It is therefore unambiguous to call the elements of S_{n} represented by evenlength words "even", and the elements represented by oddlength words "odd".
Generalizations
Parity can be generalized to Coxeter groups: one defines a length function l(v), which depends on a choice of generators (for the symmetric group, adjacent transpositions), and then the function gives a generalized sign map.
See also
 The fifteen puzzle is a classic application, though it actually involves a groupoid.
 Zolotarev's lemma
Notes
 ^ ^{a} ^{b} ^{c} ^{d} Jacobson (2009), p. 50.
 ^ Rotman (1995), p. 9, Theorem 1.6.
 ^ ^{a} ^{b} ^{c} Jacobson (2009), p. 51.
 ^ Goodman, p. 116, definition 2.4.21
 ^ Meijer & Bauer (2004), p. 72
References
 Weisstein, Eric W., "Even Permutation" from MathWorld.
 Jacobson, Nathan (2009). Basic algebra. 1 (2nd ed.). Dover. ISBN 9780486471891
 Rotman, J.J. (1995). An introduction to the theory of groups. Graduate texts in mathematics. SpringerVerlag. ISBN 9780387942858.
 Goodman, Frederick M.. Algebra: Abstract and Concrete. ISBN 9780979914201.
 Meijer, Paul Herman Ernst; Bauer, Edmond (2004). Group theory: the application to quantum mechanics. Dover classics of science and mathematics. Dover Publications. ISBN 9780486437989.
Categories: Group theory
 Permutations
 Parity
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