Perfect group

In mathematics, more specifically in group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no non-trivial abelian quotients (equivalently, its abelianization, which is the universal abelian quotient, is trivial). In symbols, a perfect group is one such that G⁽¹⁾ = G (the commutator subgroup equals the group), or equivalently one such that G^ab = {1} (its abelianization is trivial).