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
Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product.
[1] 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
, 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.
[2] 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).
A non-trivial finite perfect group that is not simple must then be an extension of at least one smaller simple non-abelian group.
Every perfect group G determines another perfect group E (its universal central extension) together with a surjection f: E → G whose kernel is in the center of E, such that f is universal with this property.
The kernel of f is called the Schur multiplier of G because it was first studied by Issai Schur in 1904; it is isomorphic to the homology group
Øystein Ore proved in 1951 that the alternating groups on five or more elements contained only commutators, and conjectured that this was so for all the finite non-abelian simple groups.
[3] A basic fact about perfect groups is Otto Grün's proposition of Grün's lemma (Grün 1935, Satz 4,[note 1] p. 3): the quotient of a perfect group by its center is centerless (has trivial center).
Proof: If G is a perfect group, let Z1 and Z2 denote the first two terms of the upper central series of G (i.e., Z1 is the center of G, and Z2/Z1 is the center of G/Z1).
If H and K are subgroups of G, denote the commutator of H and K by [H, K] and note that [Z1, G] = 1 and [Z2, G] ⊆ Z1, and consequently (the convention that [X, Y, Z] = [[X, Y], Z] is followed):
By the three subgroups lemma (or equivalently, by the Hall-Witt identity), it follows that [G, Z2] = [[G, G], Z2] = [G, G, Z2] = {1}.
Therefore, Z2 ⊆ Z1 = Z(G), and the center of the quotient group G / Z(G) is the trivial group.As a consequence, all higher centers (that is, higher terms in the upper central series) of a perfect group equal the center.
An advantage of this definition is that it admits strengthening: Especially in the field of algebraic K-theory, a group is said to be quasi-perfect if its commutator subgroup is perfect; in symbols, a quasi-perfect group is one such that G(1) = G(2) (the commutator of the commutator subgroup is the commutator subgroup), while a perfect group is one such that G(1) = G (the commutator subgroup is the whole group).