Some remarks on sharply 2-transitive groups and near-domains

TL;DR

This paper investigates the structure of sharply 2-transitive groups and near-domains in characteristic 0, proving splitting results and characterizing near-fields under certain subgroup conditions, thus answering a specific open question.

Contribution

It establishes that certain sharply 2-transitive groups and near-domains in characteristic 0 must split or be near-fields, providing new structural insights and resolving an open problem.

Findings

Sharply 2-transitive groups of characteristic 0 with abelian stabilizer subgroups split.

Near-domains of characteristic 0 with specific subgroup conditions are near-fields.

Answers an open question in the Kourovka Notebook regarding near-domains.

Abstract

A sharply 2-transitive permutation group of characteristic 0 whose point stabiliser has an abelian subgroup of finite index splits. More generally, a near-domain of characteristic 0 with a multiplicative subgroup of finite index avoiding all multipliers $d_{a,b}$ must be a near-field. In particular this answers question 12.48 b) of the Kourovka Notebook in characteristic 0.