ExampleConstruction: Literature-sourcedVerification: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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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.
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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).

[F2]

$H^{2} (G, M)$ is the quotient of normalized two-cocycles by two-coboundaries, so the zero cocycle represents its zero class (Second cohomology by factor sets).

[L1]

An extension with a specified normalized section has the cohomology class of that section's factor set, independently of the chosen normalized section (An extension determines a well-defined H^2 class).

Verification

technique · direct

1.1F1given

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

2.1F1F2L1step 1.1algebra

The explicit normalized section $s (g) = (0, g)$ is a homomorphism, so the extension splits. Its factor set is identically zero because $s (g) s (h) = s (g h)$ ; [L1] assigns the extension the class $[0]$ , which is the zero class by [F2]. This uses the displayed section and needs no choice of fibres.

3.1step 2.1∎

Therefore the zero cocycle gives the split extension.

Depends on

Used by

Nothing in the library uses this result yet.

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)