DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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Baer sum of abelian-kernel extensions

Definition

Let

$1 \to M \to E_{1} \to G \to 1, 1 \to M \to E_{2} \to G \to 1$

be two extensions inducing the same action on the abelian kernel $M$ .

Their Baer sum is the extension obtained by first forming the pullback $E_{1} \times_{G} E_{2}$ , whose kernel is naturally $M \oplus M$ , and then pushing out that kernel along the addition map $M \oplus M \to M$ , $(m, n) \mapsto m + n$ .

The resulting extension class is written $[E_{1}] + [E_{2}]$ .

Depends on

Used by

Baer sum of two factor sets Example
The Baer sum is independent of extension representatives Lemma
The Baer sum agrees with addition in H² Theorem

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)