Even and odd permutations

In mathematics, the permutations of a finite set (i. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In several fields of Mathematics the term permutation is used with different but closely related meanings In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. e. the bijective mappings from the set to itself) fall into two classes of equal size: the even permutations and the odd permutations. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property The oddness of a permutation $σ$ of the order $n$ is defined as the oddness of the number of inversions, i. e. , such pairs of indices $(i, j)$ that $1\le i<j\le n$ and $σ(i) > σ(j)$ .

The sign or signature of a permutation $σ$ is denoted sgn(σ) and defined as +1 if $σ$ is even and -1 if $σ$ . It can be explicitly expressed as

sgn(σ) = ( - 1) N (σ)

where $N (σ)$ is the number of inversions in $σ$ . Another notation for the sign of a permutation is the Levi-Civita symbol ( $\epsilon_{ijk}\ ,$ e. The Levi-Civita symbol, also called the Permutation symbol or antisymmetric symbol, is a mathematical symbol used in particular in Tensor g. ), which is also defined for non-bijective maps from the finite set to itself, with the value zero.

1 Example
2 Properties
3 Proof of connection between the sign of a permutation and the number of transpositions
4 See also
5 References

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 $\sigma=\begin{pmatrix}1&2&3&4&5\\ 3&4&5&2&1\end{pmatrix} = (1 5) (2 4) (1 3)$ There are (infinitely) 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. In several fields of Mathematics the term permutation is used with different but closely related meanings In Mathematics, a composite function represents the application of one function to the results of another

Properties

The identity permutation is an even permutation. 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. In Mathematics, the parity of an object states whether it is even or odd In informal language a transposition is a function that swaps two elements of a set

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,. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying . . ,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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function

Furthermore, we see that the even permutations form a subgroup of 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 This is the alternating group on n letters, denoted by A_n. In Mathematics, an alternating group is the group of Even permutations of a Finite set. It is the kernel of the homomorphism sgn. 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 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 Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

If n>1, then there are just as many even permutations in S_n as there are odd ones; consequently, A_n contains n!/2 permutations. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k [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. ]

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.

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

Proof of connection between the sign of a permutation and the number of transpositions

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 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 = 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.

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, . . . 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

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

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,. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations . . ,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.

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
$τ i τ i + 1 τ i = τ i + 1 τ i τ i + 1$ for all i < n-1
$τ i τ j = τ j τ i$ if |i-j| > 2. In Mathematics, one method of defining a group is by a presentation.

[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".

References

In Mathematics, Zolotarev's lemma in Number theory states that the Legendre symbol \left(\frac{a}{p}\right for

Contents

Example

Properties

Proof of connection between the sign of a permutation and the number of transpositions

See also

References