Even and odd permutations |
In s). 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.
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.
= 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=egin{bmatrix}1&2&3&4&5\ 3&4&5&2&1end{bmatrix} = (1 5) (2 4) (1 3) There are (infinitely) many other ways of writing σ as a functional 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.
= Facts =
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:
Considering the symmetric group Sn of all permutations of the set {1,..., n }, we can conclude that the map :operatorname{sgn} : S_n o {-1,1} 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 a homomorphism of the homomorphism sgn.
If n >1, then there are just as many even permutations in Sn as there odd ones; consequently, An contains factorial/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 :( 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 (group theory) must be even; the converse is not true in general.
= 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.
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 σ is a given permutation, we define an inversion pair for σ to be a pair of indices ( i , j ) such that 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 :P(x_1,ldots,x_n)=prod_{i 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 Sn represented by even-length words even , and the elements represented by odd-length words odd .
= See also =
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