CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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 finitely presented group

Statement

A finitely presented group has finitely generated Schur multiplier.

Facts & Assumptions

Given: Let $G = F / R$ have finitely many generators and finitely many defining relators.

Proof

technique · direct

1.1

Modulo $[F, R]$ , every conjugate of a defining relator has the same class as that relator. Hence the finitely many defining relators generate the abelian group $R / [F, R]$ .

givenalgebra

2.1

Hopf's formula identifies $M (G)$ with the subgroup $(R \cap [F, F]) / [F, R]$ of this finitely generated abelian group. Subgroups of finitely generated abelian groups are finitely generated, so $M (G)$ is finitely generated.

step 1.1algebra∎

Depends on

Used by

Multiplier of a finite group is finite Lemma

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 Löh, Group Cohomology (standard reference, not scraped)