I'm studying elementary level of algebra and I'm trying to prove that the commutator subgroup of a nonabelian simple group is the original group itself. It is trivially false if the group is abelian, but I can't prove it when the group is nonabelian.
The definition of the commutator subgroup of $G$ is $\{a^{-1}b^{-1}ab \mid a, b \in G\}$.