# Definition(s) > [!NOTE] Definition 1 (Sign of Permutation of Finite Set) > Let $n$ be a positive natural number and $S_{n}$ denote the $n$th [[Symmetric Groups of Finite Degree|symmetric group]]. We define $\begin{align}\text{sign}: S_{n} &\to \{ 1,-1 \} \\ \sigma & \mapsto \begin{cases} 1 & \text{if $\sigma$ is even} \\ -1 &\text{if $\sigma$ is odd}\end{cases} \end{align}$considering the [[Parity of a Permutation of n letters|parity]] of $\sigma.$ > [!Example] Example > Contents # Properties(s) [[Sign of Permutation of n Letters is a Homomorphism]] asserts that $\text{sign}$ is a homomorphism. # Application(s) **More examples**: # Bibliography