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

TL;DR

This paper provides a simpler proof, using counting arguments, to show that sharply 2-transitive subsets do not exist in certain symplectic groups, improving on previous complex proofs involving modular representation theory.

Contribution

A new, simpler proof using contradicting subsets and counting arguments for the non-existence of sharply 2-transitive sets in symplectic groups.

Findings

Non-existence of sharply 2-transitive subsets in Sp(2d,2) for specified degrees

Simplified proof avoiding modular representation theory

Use of contradicting subsets and counting arguments

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.