Another measuring argument for finite permutation groups

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 m ω,ƒ , which is the maximal value of ω (Γ)ƒ( C G (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( C G (Γ) = m ω,ƒ . Our main result, Theorem 1, states that if ℳ contains a unique maximal or minimal element Δ with respect to inclusion, then Δ g = Δ for each g ∈ G and C G (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).