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The generator rank of a finite -group
Definition
For a finite -group , the generator rank is the common size of a basis of .
This is well defined: The Frattini quotient is the largest elementary abelian quotient of a finite -group makes the quotient elementary abelian, and Finite elementary abelian -groups have bases, basis extension, and a well-defined dimension proves that it has bases and that all have the same finite size in the sense of -spanning sets, independence, and bases in an elementary abelian -group. In particular , since the trivial quotient has the empty basis.
Depends on
Used by
- A nontrivial finite p-group is cyclic exactly when d(P)=1 Corollary
- Minimal generating sets of a finite p-group have size d(P) Corollary
- The 3×3 upper-unitriangular group over a prime field has generator rank two Example
- The Frattini subgroup of (ℤ/p)ⁿ is trivial Example
- The Frattini subgroup of a nontrivial cyclic p-group Example
- The Frattini subgroups of the dihedral and quaternion groups of order eight Example
- Burnside Basis Theorem Theorem
Dependency tree · two levels
17 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
- D. A. Craven, The Theory of p-Groups, Theorem 2.28 and Definition 2.29 (standard reference, not scraped)
- K. Conrad, Generating Sets, §6 (standard reference, not scraped)