Parity of a permutation explained

In mathematics, when X is a finite set with at least two elements, the permutations of X (i.e. the bijective functions 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

\sigma 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 and .

The sign, signature, or signum 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 (ε_σ), 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

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

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 defined by

\sigma( 1 ) = 3 , \sigma( 2 ) = 4 , \sigma( 3 ) = 5 , \sigma( 4 ) = 2 , and \sigma( 5 ) = 1 . In one-line notation, this permutation is denoted 34521. It can be obtained from the identity permutation 12345 by three transpositions: first exchange the numbers 2 and 4, then exchange 3 and 5, and finally exchange 1 and 3. This shows that the given permutation σ is odd. Following the method of the cycle notation article, this could be written, composing from right to left, as\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 & 3 \end{pmatrix} \begin{pmatrix} 3 & 5 \end{pmatrix} \begin{pmatrix} 2 & 4 \end{pmatrix} . 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. An even permutation can be obtained as the composition of an even number (and only an even number) of exchanges (called transpositions) 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:

• 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, 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. 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, then there are just as many even permutations in S_n as there are odd ones; consequently, A_n contains n!/2 permutations. (The reason is that if σ is even then is odd, and if σ is odd then is even, and these two maps are inverse to each other.)

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) or (a b)(b c)(c d)(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 the same as the parity of the permutation.

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

Equivalence of the two definitions

This section presents proofs that the parity of a permutation σ can be defined in two equivalent ways:

• as the parity of the number of inversions in σ (under any ordering); or
• as the parity of the number of transpositions that σ can be decomposed to (however we choose to decompose it).

Other definitions and proofs

The parity of a permutation of

n points is also encoded in its cycle structure.

Let σ = (i₁ i₂ ... i_r+1)(j₁ j₂ ... j_s+1)...(ℓ₁ ℓ₂ ... ℓ_u+1) be the unique decomposition of σ into disjoint cycles, which can be composed in any order because they commute. A cycle involving points can always be obtained by composing k transpositions (2-cycles):

(a b c ... x y z) = (a b)(b c) ... (x y)(y z),

so call k the size of the cycle, and observe that, under this definition, transpositions are cycles of size 1. From a decomposition into m disjoint cycles we can obtain a decomposition of σ into transpositions, where k_i is the size of the ith cycle. The number is called the discriminant of σ, and can also be computed as

n minusthenumberofdisjointcyclesinthedecompositionof \sigma

if we take care to include the fixed points of σ as 1-cycles.

Suppose a transposition (a b) is applied after a permutation σ. When a and b are in different cycles of σ then

(a b)(a c _{1 c} _{2} ... c _{r)(b d} _{1 d} _{2} ... d _{s)} = (a c _{1 c} _{2} ... c _{r b d} _{1 d} _{2} ... d _{s)} ,

and if a and b are in the same cycle of σ then

(a b)(a c _{1} c _{2} ... c _{r b d} _{1 d} _{2} ... d _{s)} = (a c _{1 c} _{2} ... c _{r)(b d} _{1 d} _{2} ... d _{s)} .

In either case, it can be seen that, so the parity of N((a b)σ) will be different from the parity of N(σ).

If is an arbitrary decomposition of a permutation σ into transpositions, by applying the r transpositions

t _{1} after t₂ after ... after t_r after the identity (whose N is zero) observe that N(σ) and r have the same parity. By defining the parity of σ as the parity of N(σ), a permutation that has an even length decomposition is an even permutation and a permutation that has one odd length decomposition is an odd permutation.

Remarks:

• A careful examination of the above argument shows that, and since any decomposition of σ into cycles whose sizes sum to r can be expressed as a composition of r transpositions, the number N(σ) is the minimum possible sum of the sizes of the cycles in a decomposition of σ, including the cases in which all cycles are transpositions.
• This proof does not introduce a (possibly arbitrary) order into the set of points on which σ acts.

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
• Zolotarev's lemma

References

• • Book: Jacobson, Nathan. Nathan Jacobson. 2009. Basic algebra. 2nd. 1 . Dover. 978-0-486-47189-1.
• Book: Rotman . J.J. . An introduction to the theory of groups . Graduate texts in mathematics . 1995 . Springer-Verlag . 978-0-387-94285-8 .
• Book: Goodman . Frederick M. . Algebra: Abstract and Concrete . 978-0-9799142-0-1 .
• Book: Meijer . Paul Herman Ernst . Bauer . Edmond . Group theory: the application to quantum mechanics . Dover classics of science and mathematics . 2004 . Dover Publications . 978-0-486-43798-9 .

Notes and References

1. Jacobson (2009), p. 50.
2. Rotman (1995), [{{Google books|plainurl=y|id=lYrsiaHSHKcC|page=9|text=sgn}} p. 9, Theorem 1.6.]
3. Jacobson (2009), p. 51.
4. Goodman, [{{Google books|plainurl=y|id=l1TKk4InOQ4C|page=116|text=kernel of the sign homomorphism}} p. 116, definition 2.4.21]
5. 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]