Alphabeta MathHow 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.
This page uses the coarse five-type O'Nan-Scott convention
Modern accounts often use eight types: HA (affine), HS (holomorph simple), HC (holomorph compound), AS (almost simple), PA (product action), SD (simple diagonal), CD (compound diagonal), and TW (twisted wreath). Relative to the five labels in Affine, almost simple, diagonal, product action, and twisted wreath types, the correspondence with the Liebeck–Praeger–Saxl convention is:
| Five-type branch | Modern types |
|---|---|
| Affine (I) | HA |
| Almost simple (II) | AS |
| Simple diagonal (III(a)) | SD, HS |
| Product action (III(b)) | PA, CD, HC |
| Twisted wreath (III(c)) | TW |
In III(a), the one-minimal-normal-subgroup case is SD and the two-regular-minimal-normal-subgroup case is HS. In III(b), an almost-simple component gives PA; a simple-diagonal component gives CD or HC, according to whether there is one minimal normal subgroup or two. Thus HC and CD belong to the coarse product-action branch, not the simple-diagonal branch. This is a terminology comparison, not a proof that the five branches exhaust all finite primitive groups or a substitute for the twisted-wreath construction.
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Dependency tree · two levels
7 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
- M. W. Liebeck, C. E. Praeger and J. Saxl, On the O'Nan-Scott theorem, Section 1 (standard reference, not scraped)
- S. D. Smith, finite simple groups text, Remark 6.1.4, comparison table (standard reference, not scraped)
- Leonard H. Soicher, Primitive permutation groups (standard reference, not scraped)