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