Superperfect group

A group G is superperfect when its first two integral homology groups vanish:

H₁(G; ℤ) = 0    and    H₂(G; ℤ) = 0.

Equivalently, G is perfect (G = [G, G], which is exactly H₁ = 0) and its Schur multiplier M(G) = H₂(G; ℤ) is trivial.

Equivalent conditions

For a finite group G, all of the following say the same thing:

• H₁(G; ℤ) = H₂(G; ℤ) = 0.
• G is perfect and M(G) = {e}.
• G is its own universal central extension: the universal cover of a perfect group has kernel M(G), so it equals G exactly when M(G) is trivial.
• The only perfect central extension of G is G itself — no bigger perfect group sits over G with central kernel.
• Writing G = F/R for a free group F, Hopf's formula M(G) = (R ∩ [F, F]) / [F, R] gives R ∩ [F, F] = [F, R], together with R [F, F] = F for perfectness.

Key facts

• Superperfect ⇒ perfect. The converse fails: A₅ is perfect with M(A₅) = C₂.
• The trivial group is superperfect. The smallest non-trivial one is SL(2,𝔽5), of order 120 — the only perfect group of smaller order is A₅, and its multiplier is non-trivial.
• Among the covers of a non-abelian simple group S, exactly one is superperfect: the universal cover Ŝ, with Z(Ŝ) ≅ M(S). So every quasisimple group G is superperfect iff G ≅ Ŝ for S = G/Z(G).
• A non-abelian simple group is superperfect exactly when it is its own universal cover, i.e. M(S) = {e}.
• A direct product of superperfect groups is superperfect: the Künneth formula M(A × B) = M(A) × M(B) × (Aᵃᵇ ⊗ Bᵃᵇ) loses its last term when A, B are perfect.
• A finite group with a balanced presentation (equally many generators and relators) has trivial Schur multiplier, so a perfect one is superperfect. SL(2,𝔽5) = ⟨s, t | s³ = t⁵ = (st)²⟩ is balanced.
• Topologically, G is superperfect iff Quillen's plus construction BG⁺ is 2-connected — π₁(BG⁺) = H₁(G) and π₂(BG⁺) ≅ H₂(G).

Superperfect, but not simple

Superperfect and simple

A simple group is superperfect precisely when it has no proper cover:

By contrast A₅, PSL(2,𝔽7), A₆, PSL(2,𝔽11) and PSL(2,𝔽13) are simple but not superperfect: each has a non-trivial multiplier, hence a proper cover.

Non-examples

• Any group with a non-trivial abelian quotient — every non-trivial solvable group, in particular. Already H₁ ≠ 0.
• A₅, and more generally any simple group with M(S) ≠ {e}.
• C₂ × C₂: H₁ = C₂ × C₂ and H₂ = C₂ both non-zero.
• 2⁴:A₅, of order 960 — perfect, but its multiplier is non-trivial.

Why not "acyclic"?

A group is acyclic when every reduced homology group vanishes: Hₙ(G; ℤ) = 0 for all n ≥ 1. No non-trivial finite group is acyclic. For any prime p dividing |G|, the mod-p cohomology ring H*(G; 𝔽ₚ) is a finitely generated 𝔽ₚ-algebra whose Krull dimension is the p-rank of G, hence at least 1; so it is infinite-dimensional and Hₙ(G; ℤ) ≠ 0 for infinitely many n. Acyclic groups exist, but they are all infinite.

Superperfect is therefore the strongest vanishing a finite group can satisfy in the form "the first k homology groups are zero" — beyond k = 2 the classes would keep shrinking without ever reaching acyclicity.

In the database

The perfect group list flags each group's superperfect status. Deciding it needs the Schur multiplier, which is not yet known for every perfect group; where it is unknown the list shows ? rather than a yes or a no.

See also

• Perfect group — H₁ = 0 alone.
• Homology — H₁ = Gᵃᵇ and H₂ = M(G).
• Quasisimple group — universal covers and the multiplier.
• Simple group
• Extension — central extensions classified by M(G).
• Wikipedia: Superperfect group
• Wikipedia: Acyclic group
• Wikipedia: Homology sphere — SL(2,𝔽5) is the fundamental group of the Poincaré sphere, and the only finite group that is the fundamental group of a homology 3-sphere.