In mathematics, more specifically in the area of abstract algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelianquotients. In symbols, a perfect group is one such that G = G, or equivalently one such that Gab = .
Examples
The smallest perfect group is the alternating groupA5. More generally, any non-abeliansimple group is perfect since the commutator subgroup is a normal subgroup with abelian quotient. Conversely, a perfect group need not be simple; for example, the special linear group over the field with 5 elements, SL is perfect but not simple. The direct product of any 2 simple groups is perfect but not simple; the commutator of 2 elements =. Since commutators in each simple group form a generating set, pairs of commutators form a generating set of the direct product. More generally, a quasisimple group which is a non-trivial extension is perfect but not simple; this includes all the insoluble non-simple finite special linear groups SL as extensions of the projective special linear groupPSL is an extension of PSL. Similarly, the special linear group over the real and complex numbers is perfect, but the general linear group GL is never perfect, 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, 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, see. In fact, for n ≥ 5 the alternating group An is perfect but not superperfect, with H2 = Z/2 for n ≥ 8. Any quotient of a perfect group is perfect. A non-trivial finite perfect group which 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. Every perfect group G determines another perfect group E 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 Schur in 1904; it is isomorphic to the homology groupH2. In the plus construction of algebraic K-theory, if we consider the group for a commutative ring, then the subgroup of elementary matrices forms a perfect subgroup.
Ore's conjecture
As the commutator subgroup is generated by commutators, a perfect group may contain elements that are products of commutators but not themselves commutators. Øystein Ore proved in 1951 that the alternating groups on five or more elements contained only commutators, and made the conjecture that this was so for all the finite non-abelian simple groups. Ore's conjecture was finally proven in 2008. The proof relies on the classification theorem.
Grün's lemma
A basic fact about perfect groups is Grün's lemma from : the quotient of a perfect group by its center is centerless.
Proof: If G is a perfect group, let Z1 and Z2 denote the first two terms of the upper central series of G. If H and K are subgroups of G, denote the commutator of H and K by and note that = 1 and ⊆ Z1, and consequently : By the three subgroups lemma, it follows that = G, G], Z2] = =. Therefore, Z2 ⊆ Z1 = Z, and the center of the quotient groupG ⁄ Z is the trivial group.
As a consequence, all higher centers of a perfect group equal the center.
Group homology
In terms of group homology, a perfect group is precisely one whose first homology group vanishes: H1 = 0, as the first homology group of a group is exactly the abelianization of the group, and perfect means trivial abelianization. An advantage of this definition is that it admits strengthening:
An acyclic group is one all of whose homology groups vanish
Quasi-perfect group
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 = G, while a perfect group is one such that G = G. See and.
