ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Quaternion and dihedral central extension classes

Example

The groups $D_{8}$ and $Q_{8}$ both fit into central extensions

$1 \to C_{2} \to E \to C_{2} \times C_{2} \to 1,$

but they determine distinct classes in $H^{2} (C_{2} \times C_{2}, C_{2})$ .

Facts & Assumptions

Given: The dihedral group $D_{8}$ and the quaternion group $Q_{8}$ .

[L1]

Central extensions are classified by $H^{2}$ with trivial action (Central extensions are classified by H^2 with trivial action).

Verification

technique · direct

1.1

In both $D_{8}$ and $Q_{8}$ , the center has order $2$ , and quotienting by it gives $C_{2} \times C_{2}$ . So each group defines a central extension of $C_{2} \times C_{2}$ by $C_{2}$ .

givenalgebra

2.1

The two extensions are not equivalent because $D_{8}$ contains five involutions, while $Q_{8}$ contains only one. An extension equivalence would be an isomorphism of middle groups preserving the kernel and quotient data, but the middle groups are not isomorphic.

step 1.1algebra

3.1

Therefore [L1] assigns distinct classes in $H^{2} (C_{2} \times C_{2}, C_{2})$ to the dihedral and quaternion central extensions.

L1step 2.1∎

Depends on

Central extensions are classified by H^2 with trivial action

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 Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)