Finite transitive permutation groups with only small normaliser orbits

Abstract

We study finite transitive permutation groups $G⩽\mathrm{Sym}(\Omega )$ such that all orbits of the conjugation action on $G$ of the normaliser of $G$ in $\mathrm{Sym}(\Omega )$ have size bounded by some constant. Our results extend recent results, due to the first author, on finite abstract groups $G$ such that all orbits of the natural action of the automorphism group $\mathrm{Aut}(G)$ on $G$ have size bounded by some constant.

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