Perfect group

In

, 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

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.

More generally, a

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.^[1]

Every

). 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.

Every perfect group G determines another perfect group E (its

is in the center of E, such that f is universal with this property. The kernel of f is called the

homology group

${\displaystyle H_{2}(G)}$.

In the plus construction of algebraic K-theory, if we consider the group ${\displaystyle \operatorname {GL} (A)={\text{colim}}\operatorname {GL} _{n}(A)}$ for a commutative ring ${\displaystyle A}$, then the subgroup of elementary matrices ${\displaystyle E(R)}$ 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 conjectured 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.^[2]

Grün's lemma

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):

${\displaystyle [Z_{2},G,G]=[[Z_{2},G],G]\subseteq [Z_{1},G]=1}$

${\displaystyle [G,Z_{2},G]=[[G,Z_{2}],G]=[[Z_{2},G],G]\subseteq [Z_{1},G]=1.}$

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

) of a perfect group equal the center.

Group homology

In terms of

, 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:

• A superperfect group is one whose first two homology groups vanish: ${\displaystyle H_{1}(G,\mathbb {Z} )=H_{2}(G,\mathbb {Z} )=0}$.
• An
  acyclic group
  is one all of whose (reduced) homology groups vanish ${\displaystyle {\tilde {H}}_{i}(G;\mathbb {Z} )=0.}$ (This is equivalent to all homology groups other than ${\displaystyle H_{0}}$ 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 (Karoubi 1973, pp. 301–411) and (Inassaridze 1995, p. 76).

Notes

References

• Berrick, A. Jon; Hillman, Jonathan A. (2003), "Perfect and acyclic subgroups of finitely presentable groups",
  S2CID 30232002
• Grün, Otto (1935), "Beiträge zur Gruppentheorie. I.",
  Zbl 0012.34102
• Inassaridze, Hvedri (1995), Algebraic K-theory, Mathematics and its Applications, vol. 311, Dordrecht: Kluwer Academic Publishers Group,
  MR 1368402
• Karoubi, Max (1973), Périodicité de la K-théorie hermitienne, Hermitian K-Theory and Geometric Applications, Lecture Notes in Math., vol. 343, Springer-Verlag
• Rose, John S. (1994), A Course in Group Theory, New York: Dover Publications, Inc., p. 61,
  MR 1298629

External links

• Weisstein, Eric W. "Perfect Group". MathWorld.
• Weisstein, Eric W. "Grün's lemma". MathWorld.