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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The zero H^2 class is equivalent to splitting

Statement

Assume the Axiom of Choice (The Axiom of Choice). 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: The Axiom of Choice and an extension $1 \to M \to E \to G \to 1$ inducing the given action.

[A1]

Choice supplies the normalized sections required by the classification [L1] for arbitrary quotient surjections.

[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

1.1A1L1givenalgebra

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] under [A1].

1.2A1L1L2given

Conversely, if the class of $E$ is $0$ , then [L1] under [A1] says that $E$ is equivalent to the extension attached to the zero cocycle, namely the semidirect product $M ⋊ G$ . By [L2], that extension splits.

2.1L2step 1.1step 1.2∎

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

Depends on

Used by

Central extensions are classified by H² with trivial action Corollary

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)