Editing Permutation (section)

===Parity of a permutation===
{{main|Parity of a permutation}}

Every permutation of a finite set can be expressed as the product of transpositions.<ref>{{harvnb|Hall|1959|p=60}}</ref>
Although many such expressions for a given permutation may exist, either they all contain an even number of transpositions or they all contain an odd number of transpositions. Thus all permutations can be classified as [[Even and odd permutations|even or odd]] depending on this number.

This result can be extended so as to assign a ''sign'', written <math>\operatorname{sgn}\sigma</math>, to each permutation. <math>\operatorname{sgn}\sigma = +1</math> if <math>\sigma</math> is even and <math>\operatorname{sgn}\sigma = -1</math> if <math>\sigma</math> is odd. Then for two permutations <math>\sigma</math> and <math>\pi</math>

: <math>\operatorname{sgn}(\sigma\pi) = \operatorname{sgn}\sigma\cdot\operatorname{sgn}\pi.</math>

It follows that <math>\operatorname{sgn}\left(\sigma\sigma^{-1}\right) = +1.</math>

The sign of a permutation is equal to the determinant of its permutation matrix (below).