3-nets realizing a group in a projective plane

TL;DR

This paper classifies all 3-nets in a projective plane over an algebraically closed field of characteristic zero that realize finite groups, identifying known infinite families and unique sporadic examples, with extensions to positive characteristic.

Contribution

It provides a complete classification of 3-nets realizing finite groups in PG(2,K), including known families and sporadic cases, and extends results to positive characteristic fields.

Findings

Classified all 3-nets realizing finite groups in PG(2,K).

Identified infinite families from plane cubics and dihedral groups.

Found unique sporadic examples for quaternion groups.

Abstract

In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds true in characteristic p>0 apart from three possible exceptions Alt_4, Sym_4 and Alt_5.