ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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 natural action of $A_{n}$ is almost simple type

Example

For $n \geq 5$ , the natural action of $A_{n}$ on ${1, \dots, n}$ is of almost simple type.

Facts & Assumptions

Given: An integer $n \geq 5$ and the natural action of $A_{n}$ on ${1, \dots, n}$ .

[L1]

A finite group is almost simple when it lies between a nonabelian simple group and its automorphism group (Almost simple finite groups).

[A1]

For $n \geq 5$ , the alternating group $A_{n}$ is nonabelian simple.

Verification

technique · direct

1.1

The acting group is $A_{n}$ itself, and [A1] makes it a nonabelian simple group. Thus $A_{n} \leq A_{n} \leq Aut (A_{n})$ , so [L1] shows that $A_{n}$ is almost simple.

A1L1

2.1

In the natural primitive action, the socle is therefore $A_{n}$ itself, and this is the almost-simple branch of the O'Nan-Scott dictionary.

step 1.1∎

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

Leonard H. Soicher, Primitive permutation groups (standard reference, not scraped)

The natural action of AnAn​ is almost simple type