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

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Hopf formula for the Schur multiplier

Statement

For G=F/R with F free, M(G)≅(R∩[F,F])/[F,R].

Proof

Given: Let $G = F / R$ with $F$ free.

1.1

In the low-degree exact sequence, the kernel of $R / [F, R] \to F_{a b}$ is $(R \cap [F, F]) / [F, R]$ .

given

2.1

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

step 1.1algebra∎

Depends on

Used by

Hopf formula is presentation-independent Corollary
Kernel of the universal central extension Corollary
Multiplier of a finitely presented group Corollary
Hopf formula from a one-relator presentation Example
Multiplier of a cyclic group Proposition
Multiplier of a free group Proposition
Existence of Schur covering groups Theorem

Dependency tree · two levels

6 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)