Theorem 2

Let E be a finite set. There exists one and only one homomorphism $\varepsilon$ from $\mathfrak{S}_n$ to the multiplicative group $\{-1, +1\}$ such that $\varepsilon(\tau) = -1$ for every transposition $\tau$.

The uniqueness follows from Proposition 8. We show the existence. By transporting the structure, it may be assumed that $E = \{1, n\}$. Using the above notation, let $\varepsilon(\sigma) = (-1)^{\nu(\sigma)}$. Then (Lemma 2)

On the other hand,

As $p \neq -p$, it follows that $\varepsilon(\sigma \sigma') = \varepsilon(\sigma) \varepsilon(\sigma')$ and thus $\varepsilon$ is a homomorphism. We now show that, for every transposition $\tau$, $\varepsilon(\tau) = -1$. $\nu(\tau_{n-1, n}) = 1$, whence $\varepsilon(\tau_{n-1, n}) = -1$. As every transposition $\tau$ is conjugate to $\tau_{n-1, n}$ and the group $\{-1, +1\}$ is commutative, $\varepsilon(\tau) = \varepsilon(\tau_{n-1, n}) = -1$.