# Statement(s) > [!NOTE] Statement 1 (Sign is a homomorphism) > Let $n\in \mathbb{N}$ and $S_{n}$ denote the $n$th [[Symmetric Groups of Finite Degree|symmetric group]]. Then [[Sign of Permutation of n Letters|sign]] defined by $\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}$is a [[Homomorphism|homomorphism]]. **Remark**: It is easy to check that $\{ 1,-1 \}$ is indeed a multiplicative group. # Proof(s) **Proof of statement 1:** ... $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography