TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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Hopf formula for the Schur multiplier

Statement

Assume the Axiom of Dependent Choice and supplied projective-resolution data for group homology. For $G = F / R$ with $F$ free, $M (G) ≅ (R \cap [F, F]) / [F, R]$ .

Proof

Given: Dependent choice, the supplied group-homology resolution data, and a free presentation $G = F / R$ .

1.1givenalgebra

Under the stated hypotheses, Low-degree sequence of a free presentation gives an injection $H_{2} (G; Z) \to R / [F, R]$ whose image is the kernel of $R / [F, R] \to F_{a b}$ . This kernel is $(R \cap [F, F]) / [F, R]$ , since the map sends $r [F, R]$ to $r [F, F]$ .

2.1step 1.1algebra∎

Exactness identifies that kernel with $H_{2} (G; Z) = M (G)$ by Schur multiplier, giving the stated isomorphism.

Depends on

Used by

Multiplier of a finitely presented group Corollary
Multiplier of a cyclic group Proposition
Multiplier of a free group Proposition
Existence of Schur covering groups Theorem
Multiplier of an abelian group Theorem

Dependency tree · two levels

22 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)