Sharply 2-Transitive Linear Groups

Abstract

A group Γ is sharply 2-transitive if it admits a faithful permutation representation that is transitive and free on pairs of distinct points. Conjecturally, for all such groups there exists a near-field N (i.e. a skew field that is distributive only from the left, see Definition 2) such that ⁠. This is well known in the finite case. We prove this conjecture when Γ<GLn(F) is a linear group, where F is any field with char(F)≠2 and that p−char(Γ)≠2 (see Definition 2.2).