Parity of a permutation

Permutations of 4 elements

Odd permutations have a green or orange background. The numbers in the right column are the inversion numbers (sequence A034968 in OEIS), which have the same parity as the 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 Levi-Civita symbol ($\epsilon_\sigma$), which is defined for all maps from X to X, and has value zero for non-bijective 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 well-defined.^[1]

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

$\sigma=\begin{pmatrix}1&2&3&4&5\\ 3&4&5&2&1\end{pmatrix} = \begin{pmatrix}1&3&5\end{pmatrix} \begin{pmatrix}2&4\end{pmatrix} = \begin{pmatrix}1&5\end{pmatrix} \begin{pmatrix}1&3\end{pmatrix} \begin{pmatrix}2&4\end{pmatrix}.$

There are many other ways of writing σ as a composition of transpositions, for instance

$\sigma=(2 3) (1 2) (2 4) (3 5) (4 5),\;$

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

$\operatorname{sgn} : S_n \to \{-1,1\}$

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 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 converse is not true in general.

Equivalence of the two definitions

Proof 1

Every permutation can be produced by a sequence of transpositions (2-element 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 = T₁ T₂ ... T_k

σ = Q₁ Q₂ ... 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 T₁...T_k and Q₁...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 T₁ with σ. 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, ... 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

$P(x_1,\ldots,x_n)=\prod_{i<j} (x_i - x_j)\;$

So for instance in the case n = 3, we have

$P(x_1, x_2, x_3) = (x_1 - x_2)(x_2 - x_3)(x_1 - x_3)\;$

Now for a given permutation σ of the numbers {1,...,n}, we define

$\operatorname{sgn}(\sigma)=\frac{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}{P(x_1,\ldots,x_n)}$

Since the polynomial $P(x_{\sigma(1)},\dots,x_{\sigma(n)})$ has the same factors as $P(x_1,\dots,x_n)$ 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

$\operatorname{sgn}(\sigma\tau) = \frac{P(x_{\sigma(\tau(1))},\ldots,x_{\sigma(\tau(n))})}{P(x_1,\ldots,x_n)}$

$= \frac{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}{P(x_1,\ldots,x_n)} \cdot \frac{P(x_{\sigma(\tau(1))},\ldots, x_{\sigma(\tau(n))})}{P(x_{\sigma(1)},\ldots,x_{\sigma(n)})}$

$= \operatorname{sgn}(\sigma)\cdot\operatorname{sgn}(\tau)$

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 $\tau_1,\dots,\tau_{n-1}$ and relations

• $\tau_i^2 = 1$  for all i
• $\tau_i^{}\tau_{i+1}\tau_i = \tau_{i+1}\tau_i\tau_{i+1}$   for all i < n − 1
• $\tau_i^{}\tau_j = \tau_j\tau_i$   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".

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 $v \mapsto (-1)^{l(v)}$ gives a generalized sign map.

See also

• The fifteen puzzle is a classic application, though it actually involves a groupoid.
• Zolotarev's lemma

Notes

1. ^ ^{a b c d} Jacobson (2009), p. 50.
2. ^ Rotman (1995), p. 9, Theorem 1.6.
3. ^ ^{a b c} Jacobson (2009), p. 51.
4. ^ Goodman, p. 116, definition 2.4.21
5. ^ Meijer & Bauer (2004), p. 72

References

• Weisstein, Eric W., "Even Permutation" from MathWorld.
• Jacobson, Nathan (2009). Basic algebra. 1 (2nd ed.). Dover. ISBN 978-0-486-47189-1 
• Rotman, J.J. (1995). An introduction to the theory of groups. Graduate texts in mathematics. Springer-Verlag. 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.