Definition:Parity of Permutation

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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:

$\map \sgn \rho = 1$

where $\sgn$ denotes the sign function.

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 $-1$.

Also see