False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Every group has a universal central extension

Statement

Every group has a universal central extension.

Facts & Assumptions

Given: Take the nontrivial abelian group $C_{2}$ .

Refutation

technique · contradiction

1.1givenassume-contraalgebra

Suppose a universal central extension $u : U ↠ C_{2}$ exists. The split extension $C_{2} \times C_{2} \to C_{2}$ is central. There are two maps over $C_{2}$ from $U$ to it: $x \mapsto (u (x), 0)$ and $x \mapsto (u (x), u (x))$ . They are distinct because $u$ is surjective and $C_{2}$ is nontrivial.

2.1step 1.1discharge-contradiction∎

This contradicts the uniqueness required of a universal central extension. Thus $C_{2}$ is a counterexample to the stated assertion, independently of any choice premise.

Depends on

Universal central extension

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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)