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

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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 $Z$ .

[F1]

The zero-cocycle twisted product has multiplication $(m, g) (n, h) = (m + g \cdot n, g h)$ and canonical normalized section $g \mapsto (0, g)$ (Twisted product extension from a two-cocycle).

[F2]

The zero cocycle represents the zero element of $H^{2} (G, M)$ (Second cohomology by factor sets).

[L1]

An extension with a specified normalized section has the class of its factor set (An extension determines a well-defined H^2 class).

Refutation

technique · direct

1.1F1givenalgebra

For the inversion action of $C_{2}$ on $Z$ , [F1] constructs the split semidirect product $Z ⋊ C_{2}$ . Write $e$ for the identity and $τ$ for the nontrivial involution of $C_{2}$ . With $r = (1, e)$ and $s = (0, τ)$ , the semidirect-product law gives $s r s^{- 1} = r^{- 1} \neq r$ . Thus this extension's middle group is nonabelian, whereas the direct product $Z \times C_{2}$ is abelian. They are not isomorphic as groups, hence not equivalent as extensions.

2.1F1F2L1step 1.1

The canonical section $s (g) = (0, g)$ in [F1] has factor set zero, so [L1] assigns this extension the class $[0] = 0$ by [F2]. Since step 1.1 shows it is not the direct product, the zero class is not confined to direct products. This explicit section needs no Choice premise.

3.1step 2.1∎

Hence the statement is false.

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)