ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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.

Affine general linear groups are doubly transitive

Example

Let $V$ be a finite-dimensional vector space over a finite field. The affine general linear group $AGL (V) : = {x \mapsto A x + b : A \in GL (V), b \in V}$ acts doubly transitively on $V$ . Hence this action is primitive.

Facts & Assumptions

Given: A finite-dimensional vector space $V$ over a finite field.

[L1]

A $2$ -transitive action carries any ordered pair of distinct points to any other such pair (k-transitive and k-homogeneous actions).

[L2]

Every doubly transitive action is primitive (Every doubly transitive action is primitive).

Verification

technique · direct

1.1

Let $(x_{1}, x_{2})$ and $(y_{1}, y_{2})$ be ordered pairs of distinct points of $V$ . Then $x_{2} - x_{1}$ and $y_{2} - y_{1}$ are nonzero vectors, so there is some invertible linear map $A$ with $A (x_{2} - x_{1}) = y_{2} - y_{1}$ .

givenchoose

2.1

Define $b : = y_{1} - A x_{1}$ and $g (x) : = A x + b$ . Then $g (x_{1}) = y_{1}$ and $g (x_{2}) = A x_{2} + b = A (x_{2} - x_{1}) + y_{1} = y_{2}$ . So $AGL (V)$ is doubly transitive by [L1].

L1step 1.1construct

3.1

By [L2], the action is primitive.

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)