characterization of signature of a permutation

The signaturePlanetmathPlanetmathPlanetmath of a permutationMathworldPlanetmath is well-defined, as proved in the article. This note characterizes odd permutations.

Theorem 1.

A permutation σ is odd if and only if the number of even-order cycles in its cycle type is odd.

Thus, for example, this theoremMathworldPlanetmath asserts that (123) is an even permutation, since it has zero even-order cycles, while (12)(345) is odd, since it has precisely one even-order cycle.


Note that the function taking a permutation to its signature is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from Sn/2, and we thus get the following multiplicationPlanetmathPlanetmath rules for even and odd permutations:

(even)(even) = (odd)(odd) = (even)
(even)(odd) = (odd)(even) = (odd)

Note that we can represent a single cycle as a product of transpositionsMathworldPlanetmath:


and that therefore an even-length cycle is odd (since it is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to an odd numberMathworldPlanetmathPlanetmath of transpositions) while an odd-length cycle is even.

By the multiplication rules above, then, a given permutation is odd if and only if the product of the signs of its cycles is odd, which happens if and only if there are an odd number of cycles whose sign is odd, which happens if and only if there are an odd number of cycles of even length. ∎

Title characterizationMathworldPlanetmath of signature of a permutation
Canonical name CharacterizationOfSignatureOfAPermutation
Date of creation 2013-03-22 17:16:49
Last modified on 2013-03-22 17:16:49
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Theorem
Classification msc 03-00
Classification msc 05A05
Classification msc 20B99