TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

H^2 classifies extensions with fixed abelian kernel action

Statement

For a fixed group $G$ and abelian $G$ -module $M$ , the set of equivalence classes of extensions of $G$ by $M$ inducing the given action is in natural bijection with $H^{2} (G, M)$ .

Facts & Assumptions

Given: A group $G$ and an abelian $G$ -module $M$ .

[L1]

Every such extension determines a well-defined class in $H^{2} (G, M)$ (An extension determines a well-defined H^2 class).

[L2]

Cohomologous two-cocycles give equivalent twisted-product extensions, and equivalent twisted products have cohomologous cocycles (Cohomologous two-cocycles give equivalent extensions).

Proof

technique · direct

1.1

Map an extension class to the cohomology class of the factor set of any normalized section. This is well defined by [L1].

L1given

2.1

Every class $[f] \in H^{2} (G, M)$ is hit: choose a normalized cocycle representative $f$ , build the twisted product $M \times_{f} G$ , and use its standard section $g \mapsto (0, g)$ . The factor set of that section is exactly $f$ , so step 1.1 sends the resulting extension to $[f]$ .

L2step 1.1choosealgebra

2.2

If an extension $1 \to M \overset{i}{\to} E \overset{π}{\to} G \to 1$ has normalized section $s$ with factor set $f_{s}$ , then $Ψ : M \times_{f_{s}} G \to E, Ψ (m, g) = i (m) s (g),$ is an extension equivalence. Its inverse sends $x \in E$ to $(i^{- 1} (x s (π (x))^{- 1}), π (x)),$ and the factor-set identity shows that $Ψ$ respects multiplication.

step 1.1givenconstructalgebra

3.1

If two extensions define the same cohomology class, then after choosing normalized sections their factor sets are cohomologous. By [L2], the corresponding twisted products are equivalent extensions. Composing those equivalences with the ones from step 2.2 shows that the original extensions are equivalent. Thus the map of step 1.1 is injective.

L2step 2.2algebra

4.1

Steps 2.1 and 3.1 show that step 1.1 is a bijection.

step 2.1step 3.1∎

Depends on

Used by

Central extensions are classified by H² with trivial action Corollary
The zero H² class is equivalent to splitting Corollary
The Cₚ² extension as a nonzero two-cocycle Example
FALSE: H² classifies extensions with arbitrary nonabelian kernel False statement
Nonabelian extension obstruction in H³ Remark
Extension-theoretic interpretation of the standard five-term exact sequence Theorem
The Baer sum agrees with addition in H² Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)