CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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Kernel of the universal central extension

Statement

Assume the Axiom of Choice and supplied projective-resolution data for group homology. For a perfect group $G = F / R$ with $F$ free, the kernel of the free-presentation universal central extension $[F, F] / [F, R] \to G$ is naturally isomorphic to $M (G) = H_{2} (G; Z)$ .

Proof

Given: Use the stated choice and resolution hypotheses and the free-presentation universal extension of a perfect group $G = F / R$ . Choice supplies the infinite free-basis lifts required by Free-presentation construction is universal and implies the dependent-choice premise of the five-term sequence.

1.1given

Its kernel is $(R \cap [F, F]) / [F, R]$ .

2.1step 1.1algebra∎

Under the stated hypotheses, the free-presentation five-term exact sequence identifies its initial $H_{2} (G; Z)$ with the kernel of $R / [F, R] \to F_{a b}$ , namely $(R \cap [F, F]) / [F, R]$ . This is $M (G)$ by definition.

Depends on

Used by

Binary icosahedral cover of A5 Example
Universal central extension groups are superperfect Proposition

Dependency tree · two levels

17 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)