LemmaStatement: 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 finite group is finite

Statement

Assume the Axiom of Choice and supplied projective-resolution data for group homology. For finite $G$ , $M (G)$ is finite abelian.

Proof

Given: The stated choice and resolution hypotheses and a finite group $G$ .

1.1given

The finite-presentation lemma gives a finite presentation of $G$ ; the finitely-presented-multiplier corollary makes $M (G)$ finitely generated under the stated hypotheses. The finite-group homology torsion lemma says $∣ G ∣$ annihilates this positive-degree integral homology group.

2.1step 1.1algebra∎

A finitely generated abelian group of bounded exponent is finite, so $M (G)$ is finite.

Depends on

Used by

Schur covering group Definition
Binary icosahedral cover of A5 Example
Existence of Schur covering groups Theorem

Dependency tree · two levels

12 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)