Perfect group

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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 Gab = {1} (its abelianization is trivial).

Examples

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. 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 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 F2, 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: A5 is perfect but not acyclic (in fact, not even superperfect), see Template:Harv. In fact, for n ≥ 5 the alternating group An is perfect but not superperfect, with H2(An, Z) = Z/2 for n ≥ 8.

Every perfect group G determines another perfect group E (its universal central extension) together with a surjection f:EG 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 H2(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: 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, Z2Z1 = Z(G), and the center of the quotient group GZ(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: H1(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:

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

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  • 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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