CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The zero H^2 class is equivalent to splitting

Statement

An extension of $G$ by the abelian $G$ -module $M$ has class $0$ in $H^{2} (G, M)$ if and only if it is equivalent to the semidirect product $M ⋊ G$ , equivalently if and only if it splits.

Facts & Assumptions

Given: An extension $1 \to M \to E \to G \to 1$ inducing the given action.

[L1]

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

[L2]

A split extension is equivalent to the semidirect product extension (A group extension splits exactly when it has a complement or a compatible semidirect-product model, and a kernel retraction forces a direct product).

Proof

technique · iff

1.1

The semidirect product $M ⋊ G$ is represented by the zero cocycle, so its class in $H^{2} (G, M)$ is $0$ . Therefore any extension equivalent to $M ⋊ G$ has class $0$ by [L1].

L1givenalgebra

1.2

Conversely, if the class of $E$ is $0$ , then [L1] says that $E$ is equivalent to the extension attached to the zero cocycle, namely the semidirect product $M ⋊ G$ . By [L2], that extension splits.

L1L2given

2.1

A split extension is equivalent to a semidirect product by [L2], so steps 1.1 and 1.2 prove all claimed equivalences.

L2step 1.1step 1.2∎

Depends on

Used by

Central extensions are classified by H² with trivial action Corollary
The split extension as the zero cocycle Example
FALSE: the zero H² class corresponds to the direct product only False statement

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)