Define the sign of a permutation $\sigma \in S_{n}$. You should show that it is well-defined, and also that it is multiplicative (in other words, that it gives a homomorphism from $S_{n}$ to $\{\pm 1\})$.

Show also that (for $n \geqslant 2$ ) this is the only surjective homomorphism from $S_{n}$ to $\{\pm 1\}$.