A group is perfect if it equals its own commutator subgroup , i.e., if every element of is a product of commutators (elements of the form ).
Equivalently: let denote the abelianization of (the target of the homomorphism that is universal among homomorphisms from to abelian groups, or the largest abelian quotient of ). Then is perfect precisely when is a trivial group, since
The alternating group? is the smallest nontrivial perfect group.