Define the sign, $\operatorname{sgn}(\sigma)$, of a permutation $\sigma \in S_{n}$ and prove that it is well defined. Show that the function $\operatorname{sgn}: S_{n} \rightarrow\{1,-1\}$ is a homomorphism.

Show that there is an injective homomorphism $\psi: G L_{2}(\mathbb{Z} / 2 \mathbb{Z}) \rightarrow S_{4}$ such that $\operatorname{sgn} \circ \psi$ is non-trivial.

Show that there is an injective homomorphism $\phi: S_{n} \rightarrow G L_{n}(\mathbb{R})$ such that $\operatorname{det}(\phi(\sigma))=\operatorname{sgn}(\sigma) .$