# Definition:Parity of Permutation

## Definition

Let $n \in \N$ be a natural number.

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\rho \in S_n$, that is, let $\rho$ be a permutation of $S_n$.

The parity of $\rho$ is defined as follows:

### Even Permutation

$\rho$ is an even permutation if and only if:

$\map \sgn \rho = 1$

### Odd Permutation

$\rho$ is an odd permutation if and only if:

$\map \sgn \rho = -1$

where $\map \sgn \rho$ denotes the sign of $\rho$.

## Also defined as

Some sources define the parity of a permutation as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines its sign: that is, as $1$ and $-1$.