ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.
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Baer sum of two factor sets

Example

Let $G = C_{2}$ and $M = C_{2}$ with trivial action. If $f$ is the normalized cocycle with $f (t, t) = 1$ , then the Baer sum of the corresponding extension with itself has class

$[f] + [f] = [0] .$

Facts & Assumptions

Given: $G = M = C_{2}$ with trivial action and the cocycle $f (t, t) = 1$ .

[F1]

The Baer sum is the extension-side operation (Baer sum of abelian-kernel extensions).

[L1]

Under the classification bijection, Baer sum agrees with addition in $H^{2}$ (The Baer sum agrees with addition in H^2).

Verification

technique · direct

1.1

Because $M = C_{2}$ is written additively, the cocycle sum satisfies $(f + f) (t, t) = 1 + 1 = 0$ . So $f + f = 0$ as a cocycle.

givenalgebra

2.1

By [L1], the Baer sum of the extension class of $f$ with itself corresponds to the cohomology class of $f + f$ , which step 1.1 identified with $0$ . Thus the self-Baer-sum is the split class.

F1L1step 1.1

3.1

So this example realizes a nontrivial class whose double is zero.

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)