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.
The number of Sylow -subgroups
Definition
For a finite group and a prime , let be the set of Sylow -subgroups (Sylow -subgroups of a finite group). Define This cardinal is defined even before existence is proved because is a subset of the finite power set of ; Sylow I: every finite group has a Sylow -subgroup later shows it is nonzero.
Depends on
Used by
- A Sylow p-subgroup is normal if and only if it is unique Corollary
- Cyclic sylow does not alone imply a normal p complement Counterexample
- Sylow p-subgroups of Aut((ℤ/p)²): nₚ=p+1 Example
- The Sylow subgroups of A₅ Example
- The Sylow subgroups of S₄ Example
- Sylow II: in a finite group every p-subgroup lies in a conjugate of any Sylow p-subgroup, and the Sylow p-subgroups form a single conjugacy class Theorem
- Sylow III: nₚ≡1 (mod p) and nₚ∣ m when |G|=pᵃ m with p∤ m Theorem
- Sylow III*: nₚ(G)=[G:N_G(P)] Theorem
Dependency tree · two levels
3 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, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)