Another measuring argument for finite permutation groups
-
Avi Goren
Abstract
Let G denote a finite group of permutations of a finite set Ω. Given an orbit function ω : Ω → ℝ and a class function ƒ : G → ℝ and their extensions to subsets of Ω and G, consider the quantity mω,ƒ, which is the maximal value of ω(Γ)ƒ(CG(Γ)) over all nontrivial subsets Γ of Ω. Denote by ℳ the set of non-empty subsets Γ of Ω which satisfy m(Γ)ƒ(CG(Γ) = mω,ƒ. Our main result, Theorem 1, states that if ℳ contains a unique maximal or minimal element Δ with respect to inclusion, then Δg = Δ for each g ∈ G and CG(Δ) is a normal subgroup of G. This result may have potential future applications. As an example of an application of Theorem 1 we prove that if either G is a simple group or it is transitive on Ω, T is a normal subset of G not containing 1, Θ is a G-invariant subset of Ω and Γ ⊆ Ω, then 
(see Theorem 6).
© Walter de Gruyter
Articles in the same Issue
- A bound for the Bredon cohomological dimension
- An obstruction to the strong relative hyperbolicity of a group
- The algebra of strand splitting. I. A braided version of Thompson's group V
- Direct products and profinite completions
- The non-abelian tensor product of polycyclic groups is polycyclic
- Profinite HNN-constructions
- A finiteness condition for verbal subgroups
- Orbital digraphs of infinite primitive permutation groups
- Another measuring argument for finite permutation groups
- The strong symmetric genus of the finite Coxeter groups
- On the injectors of finite groups
- On automorphisms of finite p-groups
Articles in the same Issue
- A bound for the Bredon cohomological dimension
- An obstruction to the strong relative hyperbolicity of a group
- The algebra of strand splitting. I. A braided version of Thompson's group V
- Direct products and profinite completions
- The non-abelian tensor product of polycyclic groups is polycyclic
- Profinite HNN-constructions
- A finiteness condition for verbal subgroups
- Orbital digraphs of infinite primitive permutation groups
- Another measuring argument for finite permutation groups
- The strong symmetric genus of the finite Coxeter groups
- On the injectors of finite groups
- On automorphisms of finite p-groups
