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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
but it is impossible to write it as a product of an even number of transpositions.
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:
From these it follows that
Considering the symmetric group Sn 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.
Furthermore, we see that the even permutations form a subgroup of Sn. This is the alternating group on n letters, denoted by An. It is the kernel of the homomorphism sgn.
If n>1, then there are just as many even permutations in Sn as there odd ones; consequently, An 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.]
A cycle is even if and only if its length is odd. This follows from formulas like
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.
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.
Every transposition can be written as a product of an odd number of transpositions of adjacent elements, e.g.
If σ is a given permutation, 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 if we compose σ with the transposition (i, i+1) of two adjacent numbers, then, 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). So any product of an odd number of transpositions of adjacent elements will have an odd value of N, and any product of an even number of transpositions of adjacent elements will have an even value of N. We can now define σ to be even if N(σ) is even, 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
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(xσ(1),...,xσ(n)) has the same factors as P(x1,...,xn) 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 presentationGroup theory In mathematics, one method of defining a group is by a presentation . One specifies a set S of generators so that every element of the group can be written as a product of some of these generators, and a set T of relations among those generat of the group Sn in terms of generators τ1,...,τn-1 and relations
[Here the generator τi 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 Sn represented by even-length words "even", and the elements represented by odd-length words "odd".
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