# characterization of signature of a permutation

###### Theorem 1.

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

Thus, for example, this theorem  asserts that $(1~{}2~{}3)$ is an even permutation, since it has zero even-order cycles, while $(1~{}2)(3~{}4~{}5)$ is odd, since it has precisely one even-order cycle.

###### Proof.
 (even)$\cdot$(even) = (odd)$\cdot$(odd) = (even) (even)$\cdot$(odd) = (odd)$\cdot$(even) = (odd)

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

 $(a_{1}~{}a_{2}~{}a_{3}~{}\ldots~{}a_{k})=(a_{1}~{}a_{k})(a_{1}~{}a_{k-1})% \ldots(a_{1}~{}2)$

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 characterization  of signature of a permutation CharacterizationOfSignatureOfAPermutation 2013-03-22 17:16:49 2013-03-22 17:16:49 rm50 (10146) rm50 (10146) 4 rm50 (10146) Theorem msc 03-00 msc 05A05 msc 20B99