False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

FALSE: the zero H^2 class corresponds to the direct product only

Statement

The zero class in $H^{2} (G, M)$ corresponds only to the direct-product extension $M \times G$ .

Facts & Assumptions

Given: The inversion action of $C_{2}$ on $Z$ .

[L1]

The zero class corresponds to split extensions, equivalently semidirect products for the fixed action (The zero H^2 class is equivalent to splitting).

Refutation

technique · direct

1.1

For the inversion action of $C_{2}$ on $Z$ , the split extension is the semidirect product $Z ⋊ C_{2}$ , not the direct product $Z \times C_{2}$ , because the nontrivial element acts by $m \mapsto - m$ .

givenalgebra

2.1

By [L1], this semidirect product already represents the zero class in $H^{2} (C_{2}, Z)$ . Since step 1.1 shows it is not the direct product, the zero class is not confined to direct products.

L1step 1.1

3.1

Hence the statement is false.

step 2.1∎

Depends on

The zero H^2 class is equivalent to splitting

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)