Relations that hold for no reason
Present G as F/R with F free. Hopf's formula computes the second homology as (R ∩ [F, F])/[F, R] — the relations that are consequences of commutativity but not of conjugation. It is independent of the presentation, and it is the Schur multiplier: the obstruction to lifting projective representations and the kernel of the universal central extension of a perfect group.
Learning objectives
- State Hopf's formula and its independence of presentation.
- Compute H2 for a small group from a presentation.
- Identify the Schur multiplier and its role.
- Describe the universal central extension of a perfect group.
Section 01Hopf's formula
where 1 → R → F → G → 1 is any presentation with F free. The right-hand side is manifestly presentation-dependent; the theorem is that the quotient is not.
[F, R] consists of relations forced by the presentation itself. R ∩ [F, F] consists of relations lying in the commutator subgroup. The quotient is the relations that are ‘accidentally’ commutators — true in G for reasons not visible in the presentation. That is precisely what H2 measures.
| Group | H2(G, ℤ) |
|---|---|
| Free group | 0 — no relations at all |
| Cyclic group of order m | 0 |
| ℤ/m × ℤ/n | ℤ/gcd(m, n) |
| Free abelian of rank n | Exterior square, of rank n(n−1)/2 |
| Alternating group An, n ≥ 8 | ℤ/2 |
| Surface group of genus g | ℤ |
Section 02The Schur multiplier
For finite G, H2(G, ℤ) is isomorphic to H2(G, ℂ×) with trivial action, which classifies central extensions by the multiplicative group of the complex numbers.
Projective representations
A projective representation is a homomorphism to PGL(V). It lifts to GL(V) exactly when its class in the Schur multiplier vanishes.
Covering groups
A finite perfect group has a unique universal central extension, the Schur cover, with kernel the multiplier.
Presentations
The multiplier bounds the number of relations needed: a group with d generators and r relations satisfies r ≥ d + rank of the multiplier − rank of Gab.
If G is perfect then H1 = 0 and a universal central extension exists and is unique, with kernel exactly H2. For non-perfect groups no universal central extension exists, and the multiplier still measures central extensions but without the clean universal property.
Section 03Relation to the lower central series
Hopf's formula sits inside a broader picture relating group homology to the lower central series γ1 = G, γn+1 = [γn, G]. Stallings' and Stammbach's theorems say that a homomorphism inducing an isomorphism on H1 and a surjection on H2 induces isomorphisms on all lower central quotients.
This is the precise sense in which H1 and H2 determine the nilpotent part of a group's structure. It is the basis for the homological approach to nilpotent groups covered later in this stream.
ReferenceFrequently asked questions
Is Hopf's formula really independent of the presentation?
Yes, and that is the theorem's content. Any two presentations give canonically isomorphic quotients, which can be proved directly or deduced from the identification with the derived functor definition.
How is H<sub>2</sub> computed in practice?
For small groups, from a presentation via Hopf's formula, or by computer algebra using the Schur multiplier algorithms in group theory systems. For families, via a resolution or the Lyndon–Hochschild–Serre spectral sequence.
What does a non-zero Schur multiplier mean physically?
In quantum mechanics, that projective representations of a symmetry group do not all lift to linear ones — which is why spin representations of the rotation group require SU(2) rather than SO(3).
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