# Perfect group

In mathematics, more specifically in the area of modern 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 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).

## Examples

The smallest (non-trivial) perfect group is the alternating group *A*_{5}. More generally, any non-abelian simple 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 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 ).

More generally, a quasisimple group (a perfect central extension of a simple group) which is a non-trivial extension (i.e., 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 *A*_{5}). 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 **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.

Every acyclic group is perfect, but the converse is not true: *A*_{5} is perfect but not acyclic (in fact, not even superperfect), see Template:Harv. In fact, for *n* ≥ 5 the alternating group *A _{n}* is perfect but not superperfect, with

*H*

_{2}(

*A*,

_{n}**Z**) =

**Z**/2 for

*n*≥ 8.

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 Schur in 1904; it is isomorphic to the
homology group *H _{2}(G)*.

## Grün's lemma

A basic fact about perfect groups is **Grün's lemma** from Template:Harv: the quotient of a perfect group by its center is centerless (has trivial center).

Proof:IfGis a perfect group, letZ_{1}andZ_{2}denote the first two terms of the upper central series ofG(i.e.,Z_{1}is the center ofG, andZ_{2}/Z_{1}is the center ofG/Z_{1}). IfHandKare subgroups ofG, denote the commutator ofHandKby [H,K] and note that [Z_{1},G] = 1 and [Z_{2},G] ⊆Z_{1}, 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,Z_{2}] = [[G,G],Z_{2}] = [G,G,Z_{2}] = {1}. Therefore,Z_{2}⊆Z_{1}=Z(G), and the center of the quotient groupG⁄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.

## Group homology

In terms of group homology, a perfect group is precisely one whose first homology group vanishes: *H*_{1}(*G*, **Z**) = 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:

- A superperfect group is one whose first two homology groups vanish:
*H*_{1}(*G*,**Z**) =*H*_{2}(*G*,**Z**) = 0. - An acyclic group is one
*all*of whose (reduced) homology groups vanish (This is equivalent to all homology groups other than*H*_{0}vanishing.)

## 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*^{(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). See Template:Harv and Template:Harv.

## Notes

## References

- A. Jon Berrick and Jonathan A. Hillman, "Perfect and acyclic subgroups of finitely presentable groups", Journal of the London Mathematical Society (2) 68 (2003), no. 3, 683–698. Template:MR
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- Karoubi, M.: Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications, Lecture Notes in Math. 343, Springer-Verlag, 1973
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