Prove $\displaystyle A_n$ is the commutator subgroup of $\displaystyle S_n$ for all $\displaystyle n > 4$.
The commutator subgroup of $\displaystyle S_n$ must be normal. But for $\displaystyle n > 4$ the only non-trivial normal subgroup of $\displaystyle S_n$ is $\displaystyle A_n$.