Group homology

The integral homology groups Hₙ(G; ℤ) are abelian invariants of G. The first two have concrete descriptions.

H₁ — the abelianization

H₁(G; ℤ) ≅ Gab = G / [G, G]

is the abelianization, the quotient by the commutator subgroup. It is the largest abelian quotient of G; H₁ = {e} exactly when G is perfect.

H₂ — the Schur multiplier

H₂(G; ℤ) ≅ M(G),

the Schur multiplier. Equivalently M(G) ≅ H²(G; ℂˣ) — a cohomology group with ℂˣ coefficients — which is why it governs the projective representations and central extensions of G: it measures how far G is from having a universal central extension. For a perfect G that extension is its universal cover, with kernel M(G).

Key facts

• H₁ and H₂ are homology (subscripts); M(G) = H₂(G; ℤ) also equals the cohomology group H²(G; ℂˣ).
• |H₁| = [G : G'] = the number of 1-dimensional complex representations.
• M(A × B) = M(A) × M(B) × (Aᵃᵇ ⊗ Bᵃᵇ) (the Künneth formula).
• M(Cₙ) = {e}; a cyclic group has no non-trivial central extension.

Examples

M(A₅) = C₂ is realized by the double cover 2·A₅ ≅ SL(2,𝔽5).

See also

• Abelianization — H₁ = Gᵃᵇ.
• Extension — central extensions classified by M(G).
• Wikipedia: Group homology
• Wikipedia: Schur multiplier