Define the signature $\epsilon(\sigma)$ of a permutation $\sigma \in S_{n}$, and show that the map $\epsilon: S_{n} \rightarrow\{-1,1\}$ is a homomorphism.

Define the alternating group $A_{n}$, and prove that it is a subgroup of $S_{n}$. Is $A_{n}$ a normal subgroup of $S_{n}$ ? Justify your answer.