In mathematics, the permutations of a finite set (i.e. the bijective mappings from the set to itself) fall into two equal classes: the even permutations and the odd permutations. An even permutation is one that can be produced by an even number of exchanges of two elements (these exchanges are called transpositions). An odd permutation is one that can be produced by an odd number of transpositions. It is a remarkable and non-trivial fact that every permutation is either even or odd, but not both. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Permutation is the arrangement of symbols or objects into distinguishable orderings. ... In mathematics, a set is called finite if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, the parity of an object refers to whether it is even or odd. ... In informal language, a transposition is a function that swaps two elements of a set. ...

The sign or signature of a permutation, with the notation sgn(σ), is defined as +1 if the permutation is even and -1 if it is odd. Another notation for it is the Levi-Civita symbol, which is also defined for non-bijective maps from the finite set to itself, with the value zero. The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in tensor calculus. ...

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 2 and 4, and finally exchange 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 (infinitely) many other ways of writing σ as a composition of transpositions, for instance Permutation is the arrangement of symbols or objects into distinguishable orderings. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

,

but it is impossible to write it as a product of an even number of transpositions.

Properties

The identity permutation is an even permutation since it can be written as (1 2)(1 2).

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 In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

that assigns to every permutation its signature is a group homomorphism. Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

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). In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics an alternating group is the group of even permutations of a finite set. ... In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...

If n>1, then there are just as many even permutations in S_n as there odd ones; 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.] The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a number n is the product of all positive integers less than or equal to n. ...

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

(a b c d e) = (a b) (a c) (a d) (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.

Every permutation of odd order must be even; the converse is not true in general. In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

Proofs that every permutation is either even or odd

Every permutation can be produced by a sequence of transpositions: 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 descompositions 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 descompositions:

σ = T1 T2 ... Tk

σ = Q1 Q2 .... Qm

where all of the T1...Tk Q1...Qk are adjacent, k-k' is even, and m-m' is even.

We define an inversion pair for σ to be a pair of indices (i,j) such that i<j and σ(i)>σ(j). Let N(σ) be the number of inversion pairs of σ.

Now compose the inverse of T1 with sigma. T1 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 T2, T3, ... Tk 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 transposition 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.

An alternative proof uses the polynomial In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

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 P(x_σ(1),...,x_σ(n)) has the same factors as P(x₁,...,x_n) 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.

A third approach uses the presentation of the group S_n in terms of generators τ₁,...,τ_n-1 and relations In mathematics, one method of defining a group is by a presentation. ...

• τ_i² = 1   for all i
• τ_iτ_i+1τ_i = τ_i+1τ_iτ_i+1   for all i < n-1
• τ_iτ_j = τ_jτ_i   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 even-length word will thus always result in an even-length word after using the relations, and similarly for odd-length words. It is therefore unambiguous to call the elements of S_n represented by even-length words "even", and the elements represented by odd-length words "odd".

See also

• The fifteen puzzle is a classic application.