\begin{definition}[Definition:Odd Permutation]
Let $n \in \N$ be a natural number.
Let $S_n$ denote the symmetric group on $n$ letters.
Let $\rho \in S_n$ be a permutation in $S_n$.
$\rho$ is an '''odd permutation''' {{iff}}:
:$\map \sgn \rho = -1$
where $\sgn $ denotes the sign function.
\end{definition}