TheoremStatement: 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.

Existence of Schur covering groups

Statement

Every finite group has a Schur covering group.

Facts & Assumptions

Given: Let $G = F / R$ be a finite presentation and put $A = R / [F, R]$ and $M = (R \cap [F, F]) / [F, R]$ .

Proof

technique · direct

1.1

The quotient $A / M ≅ R / (R \cap [F, F])$ embeds in the free abelian group $F_{a b}$ , so it is free abelian. Hence $0 \to M \to A \to A / M \to 0$ splits. Choose a complement $S / [F, R]$ to $M$ in $A$ .

givenalgebra

2.1

The extension $R / S \to F / S \to G$ is central because $[F, R] \leq S$ . Its kernel is $R / S ≅ M (G)$ by Hopf's formula, and it lies in $[F / S, F / S]$ because the chosen complement meets $M$ trivially. Thus it is a stem extension with multiplier kernel. The kernel and $G$ are finite, so $F / S$ is finite, and Schur covering group makes it a Schur cover.

step 1.1algebra∎

Depends on

Used by

All finite Schur covers are unique False statement

Dependency tree · two levels

11 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)