False statementConstruction: AI-adaptedVerification: AI-generatedprecheck 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.
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FALSE: the factor set is independent of the section as a function

Statement

Changing the section leaves the factor set unchanged as a function $G \times G \to M$ .

Facts & Assumptions

Given: The direct-product extension $0 \to Z \to Z \times C_{2} \to C_{2} \to 0$ with trivial action.

[L1]

Changing the section changes the factor set by a coboundary (Changing the section changes the factor set by a coboundary).

Refutation

technique · direct

1.1givenalgebra

Let $s (1) = (0, 1)$ and $s (t) = (0, t)$ . Its factor set is $0$ . Let $s^{'} (1) = (0, 1)$ and $s^{'} (t) = (1, t)$ . Then $s^{'} (t)^{2} = (2, 1)$ , so the new factor set satisfies $f_{s^{'}} (t, t) = 2$ .

2.1L1step 1.1

Step 1.1 shows $f_{s^{'}} \neq f_{s}$ as functions. This is consistent with [L1], which says only that the two factor sets differ by a coboundary and therefore define the same cohomology class.

3.1step 2.1∎

Hence the statement is false.

Depends on

Changing the section changes the factor set by a coboundary

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)