False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-06

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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.

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All finite simple groups are alternating or cyclic

Statement

Every finite simple group is cyclic or alternating.

Facts & Assumptions

Given: Use the simple Lie-type group $PSL (2, 7)$ from the cited CFSG family table.

[L1]

Smith's CFSG family table records $PSL (2, 7)$ as a finite simple group of Lie type.

Refutation

technique · direct

1.1

This group has order $168$ . It is nonabelian, so it is not cyclic. It is not alternating: $∣ A_{5} ∣ = 60$ , $∣ A_{6} ∣ = 360$ , and $∣ A_{n} ∣ \geq 360$ for $n \geq 6$ .

L1givenalgebra

2.1

Hence a finite simple group can be neither cyclic nor alternating.

step 1.1contradiction∎

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

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