Sign is a homomorphism

Statement

Let $n \geq 2$. The map $\operatorname{sgn} \colon S_n \to \{1, -1\}$ is a surjective group homomorphism, where $\{1, -1\}$ is a group under multiplication. That is, $\operatorname{sgn}(\sigma \tau) = \operatorname{sgn}(\sigma) \operatorname{sgn}(\tau)$ for all $\sigma, \tau \in S_n$.

Remarks

If $\sigma$ is a product of $m$ transpositions and $\tau$ is a product of $l$ transpositions, then $\sigma \tau$ is a product of $m + l$ transpositions. The map is surjective because $\operatorname{sgn}((1\ 2)) = -1$. Its kernel is the alternating group $A_n$.

Explanations

L1 human

When you do two permutations one after the other, you join their chains of swaps. The number of swaps adds, so the signs multiply. This is exactly the rule of a homomorphism.