False statementConstruction: AI-adaptedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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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.

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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]

Ordinary $H^{2} (G, M)$ here is the quotient of additive cocycle and coboundary groups for an abelian $G$ -module $M$ (Normalized two-cocycle and two-coboundary, Second cohomology by factor sets, An $R$ -linear action of $G$ on a left $R$ -module, and a $G$ -module over $R$ ).

[L2]

The published extension-classification theorem fixes an abelian $G$ -module $M$ and its action (H^2 classifies extensions with fixed abelian kernel action).

Refutation

technique · direct

1.1givenL1

For a nonabelian group $N$ , the coefficient $N$ is not an abelian group, hence is not an abelian $G$ -module of the type required by [L1]. Consequently the ordinary group $H^{2} (G, N)$ asserted in the false claim is not defined by the cited construction.

2.1step 1.1L2

The classification in [L2] concerns extensions with a fixed abelian kernel and fixed action, so it cannot supply a classification for arbitrary nonabelian $N$ . The proposed classification by the ordinary group $H^{2} (G, N)$ is therefore ill-typed and false as stated.

3.1step 2.1∎

This refutes the false claim without invoking any unproved nonabelian obstruction statement.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)