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

Abstract

Abstract. We use Müller 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 Sp(2d, 2) in their doubly-transitive actions of degrees 2 2d−1 ± 2 d−1 . The original proof by Grundhöfer and Müller was rather complicated and used some results from modular representation theory, whereas our new proof requires only simple counting arguments. Sharply 2-transitive sets in finite permutation groupsLet Ω be a finite set. A set S ⊆ Sym Ω of permutations is called sharply transitive if for each α, β ∈ Ω there is a unique g ∈ S with α g = β. We call S sharply 2-transitive on Ω if it is sharply transitive on the set of pairs (ω 1 , ω 2 ) ∈ Ω 2 with ω 1 = ω 2 .Sharply transitive and sharply 2-transitive sets of permutations correspond to Latin squares and projective planes, respectively, and are therefore relevant to the fields of combinatorics and geometry.In the 1970s, Lorimer started the classification of all finite permutation groups containing a sharply 2-transitive subset, see [6] for a summary of his work. In 1984, O'Nan continued Lorimer's research and mentions in [9] that "work on the groups Sp(2d, 2) of degrees 2 2d−1 ±2 d−1 is in progress" -these are the permutation groups our paper will mainly focus on. Later in 2009, Grundhöfer and Müller [5] extended O'Nan's character-theoretic methods and finally showed -among many non-existence results for other almost simple finite permutation groups -that there are no sharply 2-transitive sets in Sp(2d, 2) of degree 2 2d−1 ± 2 d−1 . Their complicated proof was based on results by Sastry and Sin [10] regarding characteristic 2 permutation representations for symplectic groups.In 2010, Müller and Nagy [8] introduced the so-called method of contradicting subsets and were -in addition to proving new results -also able to give simple combinatorial proofs for most of the previously known results by Lorimer, O'Nan, Grundhöfer and Müller.However, they didn't reprove the non-existence of sharply 2-transitive sets in the symplectic groups Sp(2d, 2) of degrees 2 2d−1 ± 2 d−1 . This is done in the present paper. More precisely, we construct contradicting subsets for the orthogonal groups O ± (2d, 2), d ≥ 4, in their natural actions on singular vectors -which happen to be the point stabilizers of the symplectic groups.