ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

Projective linear actions and Iwasawa's hypotheses

Example

Let $q > 3$ , let $G = {PSL}_{2} (F_{q})$ , and let $P^{1} (F_{q}) = F_{q} \cup {\infty} .$ The usual fractional linear action of $G$ on $P^{1} (F_{q})$ is doubly transitive and therefore primitive. The stabilizer of $\infty$ contains the translation subgroup $A : = {x \mapsto x + b : b \in F_{q}},$ which is abelian and normal there, and the conjugates of $A$ generate $G$ . So this action satisfies the hypotheses of Iwasawa's criterion.

Facts & Assumptions

Given: The action of ${PSL}_{2} (F_{q})$ on $P^{1} (F_{q})$ by fractional linear transformations.

[L1]

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

[L2]

Iwasawa's criterion applies to a faithful primitive action when the point stabilizer contains a nontrivial abelian normal subgroup whose conjugates generate the whole group (Iwasawa's simplicity criterion for primitive actions).

Verification

technique · direct

1.1

Fractional linear transformations send any ordered pair of distinct points of $P^{1} (F_{q})$ to any other such pair, so the action is doubly transitive and therefore primitive by [L1].

1.2

The subgroup $A$ fixes $\infty$ , is abelian under composition, and is normal in the stabilizer of $\infty$ because conjugating a translation by an affine map gives another translation.

givenalgebra

2.1

Conjugating $A$ by the inversion $x \mapsto - 1 / x$ gives the lower-unitriangular subgroup. The upper and lower unitriangular subgroups generate ${SL}_{2} (F_{q})$ by Gaussian elimination, so their projective images generate $G$ . Thus the conjugates of $A$ generate $G$ , and [L2] applies.

L2step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

P. J. Cameron, Classical Groups, Sections 2.3-2.4 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)