Pipeline-generated

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.

✓ 6 results · all verified · 3 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

The Finite Simple Group Classification Landscape — Examples

1 · Prerequisites

2 · Summary

These witnesses illustrate the named classification families and the extension-data warning; they do not construct a sporadic group or prove CFSG.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

Cyclic simple groups of prime order

Example

$C_{p}$ is finite simple for every prime $p$ .

Facts & Assumptions

Given: Let $p$ be prime.

Verification

technique · direct

1.1

The group $C_{p}$ has order $p$ , so Lagrange's theorem leaves only subgroups of orders $1$ and $p$ . Thus it has no nontrivial proper normal subgroup.

givenalgebra

2.1

Hence $C_{p}$ is finite and simple.

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-06Open item page →

A5 as the smallest nonabelian simple group

Example

$A_{5}$ is the smallest nonabelian finite simple group.

Facts & Assumptions

Given: Use $A_{n}$ is simple for every $n \geq 5$ and the cited CFSG introductory table.

[L1]

Smith's introductory table identifies $A_{5}$ as the least-order nonabelian finite simple group.

Verification

technique · direct

1.1

The dependency makes $A_{5}$ simple, and $∣ A_{5} ∣ = 5! / 2 = 60$ ; it is nonabelian. The cited source identifies it as the least-order nonabelian simple group.

L1givenalgebra

2.1

Therefore $A_{5}$ is the smallest nonabelian finite simple group.

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

PSL(2,7) and a low-rank family entry

Example

$PSL (2, 7)$ is a sourced Lie-type entry under the table's low-rank convention.

Facts & Assumptions

Given: Use Smith's Lie-type family table cited above.

[L1]

Smith's table places $PSL (2, 7)$ in the projective special linear family under its stated low-rank conventions.

Verification

technique · direct

1.1

In that table $PSL (2, 7)$ occurs in the projective special linear family, with the source's stated low-rank naming conventions.

L1given

2.1

This is precisely the claimed table-level Lie-type example; no structural construction is being inferred.

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06Open item page →

A Suzuki group family entry

Example

Suzuki groups occur in the source’s nonclassical twisted row.

Facts & Assumptions

Given: Use Smith's nonclassical twisted-family row cited above.

[L1]

Smith's nonclassical twisted-family row includes the Suzuki groups among the finite simple groups of Lie type.

Verification

technique · direct

1.1

That row explicitly includes the Suzuki family among the finite simple groups of Lie type.

L1given

2.1

Hence Suzuki groups furnish the asserted named-family entry.

step 1.1∎

ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-06Open item page →

The Mathieu groups among the sporadics

Example

$M_{11}$ , $M_{12}$ , $M_{22}$ , $M_{23}$ , and $M_{24}$ are Mathieu entries in the sporadic list.

Facts & Assumptions

Given: Use the sporadic list recorded in The twenty-six sporadic simple groups.

Verification

technique · direct

1.1

The cited list names $M_{11}, M_{12}, M_{22}, M_{23}, M_{24}$ as the five Mathieu sporadic simple groups.

given

2.1

These are therefore exactly the claimed Mathieu entries.

step 1.1∎

CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-06Open item page →

Composition factors do not determine the extension

Statement refuted

C4 and C2×C2 share composition factors but are not isomorphic.

Counterexample

Given: Use the two groups in the statement.

1.1

Their simple factors agree while their element orders differ.

given

2.1

Thus they provide the asserted counterexample.

step 1.1algebra∎

Sources

Stephen D. Smith, CFSG—A User’s Manual