Prove that the sign function on permutations is unique in the following sense. If $f$ is any function which assigns to each permutation of degree $n$ an integer, and if $f(στ) = f(σ)f(τ)$, then $f$ is identically $0$, or $f$ is identically $1$, or $f$ is the sign function.
I think this can be converted into an homomorphism applied to Symmetric group. But I don't know how to solve.
Hint: If the function $f:S_n\rightarrow {\Bbb Z}$ is a homomorphism, $f(\sigma\tau) = f(\sigma)f(\tau)$, then the image $H=f(S_n)$ must be a multiplicative subgroup of $\Bbb Z$. But the only nontrivial multiplicative subgroup of $\Bbb Z$ is $\{\pm 1\}$.