H2 classifies extensions with a given action
An extension of G by an abelian A determines an action of G on A. Fixing that action, the extensions are classified by H2(G, A): choose a set-theoretic section, measure its failure to be a homomorphism by a factor set, and check that the factor set is a 2-cocycle and that changing the section changes it by a coboundary. The zero class is the semidirect product.
Learning objectives
- Construct the factor set of an extension from a section.
- Verify the cocycle condition and the effect of changing the section.
- State the classification theorem.
- Identify central extensions and the special role of trivial action.
Section 01From extensions to cocycles
- Given 1 → A → E → G → 1 with A abelian, choose a set-theoretic section s: G → E with s(1) = 1.
- Define the action of G on A by conjugation: g·a = s(g) a s(g)−1. Independent of the section because A is abelian.
- Define f(g, h) = s(g)s(h)s(gh)−1, which lies in A since it maps to 1 in G.
- Associativity in E forces the 2-cocycle condition on f.
- Changing the section by c: G → A changes f by the coboundary of c.
- So the class [f] ∈ H2(G, A) is an invariant of the extension.
The trivial class corresponds to a factor set that is identically 1, meaning the section is a homomorphism, meaning the extension splits. So H2 = 0 says every extension with that action is a semidirect product.
Section 02Central extensions
When the action is trivial, A is central in E, and H2(G, A) with trivial coefficients classifies central extensions.
Projective representations
A projective representation of G lifts to a linear representation of a central extension. The obstruction lies in H²(G, ℂ×), the Schur multiplier.
Covering groups
SU(2) → SO(3) is a central extension with kernel of order 2, represented by the non-trivial class in H²(SO(3), ℤ/2).
Universal central extension
A perfect group has a universal central extension whose kernel is its Schur multiplier H2(G, ℤ).
Section 03The obstruction reading
More generally H2 is where obstructions live. Given an action of G on a non-abelian group with centre A, an extension realising it exists exactly when a certain class in H3(G, A) vanishes; when it does, the extensions are a torsor under H2.
H1 counts the ways a thing can be done; H2 says whether it can be done at all; H3 obstructs the H2 problem in turn. The same pattern appears in deformation theory, in gerbes and in the theory of Brauer groups.
ReferenceFrequently asked questions
Why must the kernel be abelian?
Because the conjugation action of E on A must factor through G, which requires A to act trivially on itself. For non-abelian kernels the classification involves outer actions and an H³ obstruction, and is substantially more intricate.
Is the correspondence with H² a group isomorphism?
It is a bijection of sets that becomes a group isomorphism when extensions are added by the Baer sum, exactly as for module extensions. The two constructions agree.
What is the Schur multiplier?
H2(G, ℤ), equivalently H²(G, ℂ×) for finite G. It measures the obstruction to lifting projective representations and is the kernel of the universal central extension of a perfect group.
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