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.
Maximal proper subgroups
Definition
A subgroup is maximal proper when there is no subgroup with . Equivalently, every subgroup containing is either or . The word maximal refers to inclusion among proper subgroups, not to cardinality.
Depends on
Used by
- Maximal subgroups of a finite p-group are the inverse images of Frattini hyperplanes Corollary
- The Frattini subgroup Φ(G) as the intersection of the maximal subgroups of a finite group Definition
- FALSE: the Frattini subgroup is the union of the maximal subgroups False statement
- Maximal subgroups of finite nilpotent groups are normal of prime index Theorem
Dependency tree · two levels
5 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
- Keith Conrad, The Sylow Theorems, Sections 1-2 (standard reference, not scraped)