TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Free-presentation construction is universal

Statement

For perfect G, [F,F]/[F,R]→G is universal.

Facts & Assumptions

Given: Let $E \to G$ be a central extension and lift the free generators of $F$ to $E$ .

Proof

technique · direct

1.1

The induced map from $F$ kills $[F, R]$ on commutators because the kernel of $E \to G$ is central. It therefore restricts to a map $[F, F] / [F, R] \to E$ over $G$ .

given

2.1

Two choices of lifts of the free generators differ by elements of the central kernel, so their maps agree on every commutator and hence on $[F, F]$ . Thus the descended map is independent of the lifts and is the unique map over $G$ , proving the universal property.

step 1.1algebra∎

Depends on

Used by

Kernel of the universal central extension Corollary
Universal central extension groups are superperfect Proposition

Dependency tree · two levels

9 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)