On transitive sets of derangements in primitive groups

Abstract

We construct a primitive permutation action of the Steinberg triality group ${}^3{D}_4(2)$ of degree $4064256$ and show that there are distinct points $\hat{I}\pm ,{\hat{I}}^2$ such that there is no derangement $g\in {}^3{D}_4(2)$ with $\hat{I}{\pm }^g={\hat{I}}^2$. This answers a question by John G. Thompson (Problem 8.75 in the Kourovka Notebook) in the negative.

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