ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

Central extensions of a cyclic group

Example

Let $G = C_{n} = ⟨ x ⟩$ and let $A$ be an abelian group with trivial $C_{n}$ -action. A central extension class is represented by a relation

${\tilde{x}}^{n} = a \in A,$

and changing the lift $\tilde{x}$ by $b \in A$ changes $a$ to $a + n b$ . Hence the extension classes are parametrized by $A / n A$ .

Facts & Assumptions

Given: A cyclic quotient $C_{n} = ⟨ x ⟩$ and a trivial-action abelian kernel $A$ .

[L1]

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

Verification

technique · direct

1.1

In any central extension, choose a lift $\tilde{x}$ of the generator $x$ . Since the quotient has order $n$ , the element ${\tilde{x}}^{n}$ lies in the kernel $A$ . That kernel is central, so the extension is determined by the parameter $a : = {\tilde{x}}^{n}$ .

L1givenchoosealgebra

2.1

Replacing $\tilde{x}$ by $b \tilde{x}$ with $b \in A$ changes the parameter to $(b \tilde{x})^{n} = n b + {\tilde{x}}^{n}$ because $A$ is central and written additively. Thus two parameters define equivalent extensions exactly when they differ by an element of $n A$ .

step 1.1algebra

3.1

So the central extension classes are parametrized by $A / n A$ , which is the familiar description of $H^{2} (C_{n}, A)$ from [L1].

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)