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.

The smallest (non-trivial) perfect group is the alternating group A₅. More generally, any non-abelian simple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. However, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL(2,5) (or the binary icosahedral group, which is isomorphic to it) is perfect but not simple (it has a non-trivial center containing ${\displaystyle -\!\left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)=\left({\begin{smallmatrix}4&0\\0&4\end{smallmatrix}}\right)}$).

The direct product of any two simple non-abelian groups is perfect but not simple; the commutator of two elements is [(a,b),(c,d)] = ([a,c],[b,d]). Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product.

The fundamental group of ${\displaystyle SO(3)/I_{60}}$ is a perfect group of order 120.

More generally, a quasisimple group (a perfect central extension of a simple group) that is a non-trivial extension (and therefore not a simple group itself) is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL(n,q) as extensions of the projective special linear group PSL(n,q) (SL(2,5) is an extension of PSL(2,5), which is isomorphic to A₅). Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect (except when trivial or over ${\displaystyle \mathbb {F} _{2}}$, where it equals the special linear group), as the determinant gives a non-trivial abelianization and indeed the commutator subgroup is SL.

A non-trivial perfect group, however, is necessarily not solvable; and 4 divides its order (if finite), moreover, if 8 does not divide the order, then 3 does.

Every acyclic group is perfect, but the converse is not true: A₅ is perfect but not acyclic (in fact, not even superperfect), see (Berrick & Hillman 2003). In fact, for ${\displaystyle n\geq 5}$ the alternating group ${\displaystyle A_{n}}$ is perfect but not superperfect, with ${\displaystyle H_{2}(A_{n},\mathbb {Z} )=\mathbb {Z} /2}$ for ${\displaystyle n\geq 8}$.

Any quotient of a perfect group is perfect. A non-trivial finite perfect group that is not simple must then be an extension of at least one smaller simple non-abelian group. But it can be the extension of more than one simple group. In fact, the direct product of perfect groups is also perfect.