TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-26 (gpt-6-sol)

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The Baer sum agrees with addition in H^2

Statement

Assume the Axiom of Choice (The Axiom of Choice). Under the classification bijection between extension classes and $H^{2} (G, M)$ , the Baer sum of extensions corresponds to addition of cohomology classes.

Facts & Assumptions

Given: The Axiom of Choice and two extension classes of $G$ by the abelian $G$ -module $M$ .

[A1]

Choice supplies normalized sections for the classification [L1] of arbitrary extensions.

[L1]

$H^{2} (G, M)$ classifies the extension classes (H^2 classifies extensions with fixed abelian kernel action).

[F1]

The Baer sum is defined on extension classes (Baer sum of abelian-kernel extensions).

[L2]

That operation is independent of the chosen representatives (The Baer sum is independent of extension representatives).

Proof

technique · direct

1.1A1L1F1givenchoosealgebra

Under [A1], choose cocycle representatives $f_{1}$ and $f_{2}$ for the two classes via [L1]. The corresponding twisted-product extensions have a pullback whose kernel is $M \oplus M$ , and pushing out along addition sends the pair $(f_{1}, f_{2})$ to the cocycle $f_{1} + f_{2}$ .

2.1L2step 1.1algebra

Therefore the extension class of the Baer sum corresponds to the cohomology class $[f_{1} + f_{2}] = [f_{1}] + [f_{2}]$ in $H^{2} (G, M)$ . By [L2], this description does not depend on the chosen cocycle representatives.

3.1step 2.1∎

So the classification bijection is an additive identification.

Depends on

Used by

Baer sum of two factor sets Example

Dependency tree · two levels

9 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)