False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Composition factors determine the finite group

Statement

Two finite groups with the same multiset of composition factors are isomorphic.

Facts & Assumptions

Given: Compare $C_{4}$ and $C_{2} \times C_{2}$ .

Refutation

technique · direct

1.1givenalgebra

Each has a composition series with two factors isomorphic to $C_{2}$ . However, $C_{4}$ contains an element of order four and $C_{2} \times C_{2}$ does not.

2.1step 1.1contradiction∎

The groups are therefore nonisomorphic despite having the same multiset of composition factors.

Depends on

The Jordan–Hölder theorem for groups

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

Stephen D. Smith, CFSG—A User’s Manual (standard reference, not scraped)