Moufang sets generated by translations in unitals

TL;DR

This paper classifies the groups generated by translation centers in certain unitals, showing they are isomorphic to well-known groups like SL, PSL, Suzuki, or Ree groups, and describes their action properties.

Contribution

It identifies the specific types of groups generated by translations in unitals with two center points, extending understanding of their structure and symmetry.

Findings

G is either SL(2, F_q), PSL(2, F_q), Suzuki, or Ree groups.

G induces a Moufang set on the block joining the two points.

G is semi-regular outside the special block.

Abstract

We consider unitals of order $q$ with two points which are centers of translation groups of order $q$. The group $G$ generated by these translations induces a Moufang set on the block joining the two points. We show that $G$ is either $\operatorname{SL}(2,\mathbb{F}_q)$ (as in all classical unitals and also in some non-classical examples), or $\operatorname{PSL}(2,\mathbb{F}_q)$, or a Suzuki or a Ree group. Moreover, $G$ is semi-regular outside the special block.