CorollaryStatement: Literature-sourcedProof: 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.

The finite Iwasawa criterion

Statement

Let a finite group $G$ act faithfully and primitively on $Ω$ , and fix $α \in Ω$ . Assume $G_{α}$ has a nontrivial abelian normal subgroup $A$ whose conjugates generate $G$ . If $G = [G, G]$ , then $G$ is simple.

Facts & Assumptions

Given: A finite faithful primitive action of $G$ on $Ω$ , a point $α \in Ω$ , a nontrivial abelian normal subgroup $A ⊴ G_{α}$ , the conjugates of $A$ generate $G$ , and $G = [G, G]$ .

[L1]

Under these hypotheses, every nontrivial normal subgroup of $G$ contains $[G, G]$ , and therefore a group with $G = [G, G]$ is simple (Iwasawa's simplicity criterion for primitive actions).

Proof

technique · direct

1.1

The stated hypotheses are exactly those of [L1].

2.1

Since $G = [G, G]$ , the concluding clause of [L1] applies and yields that $G$ is simple.

L1step 1.1∎

Depends on

Iwasawa's simplicity criterion for primitive actions

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)