Sharply 2-transitive linear groups

TL;DR

This paper proves a conjecture relating sharply 2-transitive linear groups to near-fields, specifically when the group is a subgroup of GL(n,F) over a field with certain characteristic restrictions.

Contribution

It establishes the conjecture for linear groups G < GL(n,F), assuming the characteristic of F and the permutational characteristic are not 2.

Findings

Proves the conjecture for linear groups under specified conditions

Shows the structure of G corresponds to near-fields in this setting

Extends known finite case results to linear groups

Abstract

A group G 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) such that G is isomorphic to the semidirect product of the multiplicative and additive groups of N. This is well known in the finite case. We prove this conjecture when G < GL(n,F) is a linear group. Here we have to assume that both the characteristic of the field F and the permutational characteristic of the group G (see Definition 2.1) are not equal to 2.