The non-existence of sharply $2$-transitive sets of permutations in $\mathrm{Sp}(2d,2)$ of degree $2^{2d-1} \pm 2^{d-1}$

Abstract

We use M\"uller and Nagy's method of contradicting subsets to give a new proof for the non-existence of sharply $2$-transitive subsets of the symplectic groups $\mathrm{Sp}(2d,2)$ in their doubly-transitive actions of degrees $2^{2d-1}\pm 2^{d-1}$. The original proof by Grundh\"ofer and M\"uller was rather complicated and used some results from modular representation theory, whereas our new proof requires only simple counting arguments.