PropositionStatement: 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.

Multiplier of a cyclic group

Statement

Assume the Axiom of Choice and supplied projective-resolution data for group homology. Then $M (C) = 0$ for every cyclic group $C$ .

Proof

Given: The stated choice and resolution hypotheses. For finite cyclic $C_{n}$ , take $F = ⟨ x ⟩$ and $R = ⟨ x^{n} ⟩$ ; the infinite cyclic group has $R = 1$ . Choice implies the dependent choice required by Hopf formula for the Schur multiplier.

1.1given

The group $F$ is abelian, hence $R \cap [F, F] = 1$ .

2.1step 1.1algebra∎

Under the stated hypotheses, Hopf's formula gives $M (C) = (R \cap [F, F]) / [F, R] = 0$ in both cases. The trivial group is also cyclic; its free presentation $F = R = 1$ gives the same result.

Depends on

Used by

Binary icosahedral cover of A5 Example
Multiplier of a cyclic group Example
Universal coefficients split naturally False statement

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)