Alphabeta MathHow statement and proof provenance work
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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
- 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 pmod p and nₚ∣ m when |G|=pᵃ m with p∤ m Theorem
- Sylow III*: nₚ(G)=[G:N_G(P)] Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 27 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)