TheoremStatement: 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.

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

Existence criterion for universal central extensions

Statement

Assume the Axiom of Choice. A group $G$ admits a universal central extension if and only if $G$ is perfect.

Facts & Assumptions

Given: The Axiom of Choice. First suppose $u : U ↠ G$ is universal. In the converse, Choice supplies simultaneous lifts of the generators of a free presentation into each central extension.

Proof

technique · direct

1.1givenalgebra

For every abelian group $A$ and homomorphism $ϕ : G \to A$ , the two maps $x \mapsto (u (x), 0)$ and $x \mapsto (u (x), ϕ (u (x)))$ from $U$ to the split central extension $G \times A \to G$ must agree by universality. Since $u$ is surjective, $ϕ = 0$ . Taking $A = G_{a b}$ and $ϕ$ the quotient map gives $G_{a b} = 0$ , so $G$ is perfect.

1.2givenconstruct

Conversely, let $G = F / R$ be perfect, using for example the free group on the underlying set of $G$ . Under Choice, Free-presentation construction is universal proves that $[F, F] / [F, R] \to G$ is a universal central extension. Its proof identifies the exact use of Choice: lifting every free generator through each central quotient.

2.1step 1.1step 1.2∎

Steps 1.1 and 1.2 prove the two implications.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)