Group action orbital

Every arc-transitive digraph is an orbital digraph

A (simple) digraph is arc-transitive iff it is the orbital digraph for some permutation group. graph

Proof

Let Γ be an arc-transitive digraph and 𝐺 =Aut⁡(Γ). Then for any (𝛼,𝛽) ∈A⁡(Γ) we have 𝐺(𝛼,𝛽) =A⁡(Γ). Therefore Γ is an orbital digraph.

Now suppose 𝐺 acts on Ω and let Δ =𝐺(𝛼,𝛽) be an orbital. Then (𝛼′,𝛽′) ∈Δ iff (𝛼′,𝛽′) =𝑔(𝛼,𝛽) for some 𝑔 ∈𝐺 ≤Aut⁡(Γ), thus Δ is arc-transitive.

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