LemmaStatement: Literature-sourcedProof: 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.

Distinct components commute

Statement

Distinct components of a finite group commute.

Facts & Assumptions

Given: Let $K$ and $L$ be distinct components of $G$ .

[L1]

Smith's component theorem says that two components of a finite group either coincide or centralize one another.

Proof

technique · direct

1.1

Smith's component theorem says that two components of a finite group either coincide or centralize one another. It applies because $K$ and $L$ are subnormal quasisimple subgroups, exactly the components of Quasisimple groups, components, and the layer.

L1given

2.1

Since $K \neq L$ , the second alternative gives $[K, L] = 1$ ; equivalently, $K$ and $L$ commute.

step 1.1∎

Depends on

Quasisimple groups, components, and the layer

Used by

The generalized Fitting subgroup contains its centralizer Theorem

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)