False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Every central extension is stem

Statement

Every central extension is a stem extension.

Facts & Assumptions

Given: Consider the split central extension $C_{2} \to C_{2} \times C_{2} \to C_{2}$ .

Refutation

technique · direct

1.1givenalgebra

Its kernel is central, but the total group is abelian, so its commutator subgroup is trivial and does not contain the nontrivial kernel.

2.1step 1.1contradiction∎

The extension is central but not stem, refuting the assertion.

Depends on

Central and stem extensions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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)