Perfect Group

• The alternating group $A_n$ is perfect for $n\ge 5$.
• The trivial group is perfect.
• A nontrivial abelian group is not perfect because its commutator subgroup is trivial.

Equivalently, $G$ is perfect if and only if its abelianization $G/[G,G]$ is trivial, or, equivalently, every homomorphism from $G$ to an abelian group is trivial. Every nonabelian simple group is therefore perfect.