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

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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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The split extension as the zero cocycle

Example

When $f = 0$ , the twisted product $M \times_{f} G$ is the semidirect product $M ⋊ G$ , so it represents the zero class in $H^{2} (G, M)$ .

Facts & Assumptions

Given: A group $G$ , an abelian $G$ -module $M$ , and the zero function $f (g, h) = 0$ .

[F1]

The twisted product uses the multiplication $(m, g) (n, h) = (m + g \cdot n + f (g, h), g h)$ (Twisted product extension from a two-cocycle).

[L1]

The zero class corresponds exactly to split extensions (The zero H^2 class is equivalent to splitting).

Verification

technique · direct

1.1

With $f = 0$ , [F1] becomes $(m, g) (n, h) = (m + g \cdot n, g h),$ which is the usual semidirect-product law on $M ⋊ G$ .

F1given

2.1

The section $g \mapsto (0, g)$ is then a homomorphism, so the extension splits. By [L1], this is precisely the zero class in $H^{2} (G, M)$ .

L1step 1.1algebra

3.1

Therefore the zero cocycle gives the split extension.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)