Prove that in $S_n$ there are an equal number of even and odd permutations.

The map $\sigma \mapsto (12)\sigma$ is a bijection and it maps even permutations to odd ones and vice-versa.

The map $\sigma \mapsto \hbox{sign}(\sigma)$ is a surjective homomorphism $S_n \to C_2$ whose kernel is the set of even permutations.