False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-05‡ rests on unproved material

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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: H^2 classifies extensions with arbitrary nonabelian kernel

Statement

The group $H^{2} (G, N)$ classifies extensions of $G$ by an arbitrary nonabelian kernel $N$ .

Facts & Assumptions

Given: A nonabelian kernel $N$ .

[L1]

The theorem on this page classifies extensions only for abelian kernels (H^2 classifies extensions with fixed abelian kernel action).

‡ [L2]

For nonabelian kernels the obstruction moves to $H^{3}$ (Nonabelian extension obstruction in H^3 ‡).

Refutation

technique · direct

1.1

The hypothesis of [L1] requires the kernel to be abelian, so it does not apply to a general nonabelian $N$ .

L1given

2.1

The boundary remark [L2] states the right replacement: nonabelian extensions are controlled by an $H^{3}$ obstruction together with an $H^{2}$ torsor when the obstruction vanishes. So a single $H^{2}$ group does not classify them.

‡ L2step 1.1

3.1

Therefore the statement is false.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)