CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (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.

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Central extensions of perfect groups

Statement

Assume the Axiom of Choice and supplied projective and injective resolution data for group (co)homology. For a perfect group $G$ and a trivial $G$ -module $A$ , equivalence classes of central extensions of $G$ by $A$ are naturally in bijection with $Hom (M (G), A)$ .

Facts & Assumptions

Given: The stated choice and resolution hypotheses, a perfect group $G$ , and an abelian group $A$ with trivial $G$ -action.

Proof

technique · direct

1.1givenalgebra

Perfectness gives $G_{a b} = 0$ , hence ${Ext}_{Z}^{1} (G_{a b}, A) = 0$ . The degree-two universal-coefficient sequence therefore identifies $H^{2} (G; A)$ naturally with $Hom (M (G), A)$ .

2.1step 1.1algebra∎

The extension-classification theorem identifies $H^{2} (G; A)$ with equivalence classes of extensions inducing the trivial action, namely central extensions. Composing the two bijections proves the claim.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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 Löh, Group Cohomology (standard reference, not scraped)