Sharply 2-transitive groups of finite Morley rank

TL;DR

This paper classifies sharply 2-transitive groups of finite Morley rank, showing they split or are affine transformations of algebraically closed fields depending on characteristic, with additional results in characteristic 0.

Contribution

It provides a comprehensive classification of sharply 2-transitive groups of finite Morley rank across different characteristics, extending known results.

Findings

Groups of characteristic 2 split

Groups of characteristic ≠ 2 are affine transformations of algebraically closed fields

Characteristic 3 groups are affine transformations of algebraically closed fields of characteristic 3

Abstract

A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the group of affine transformations of an algebraically closed field. In particular, a sharply 2-transitive permutation group of finite Morley rank of characteristic 3 is the group of affine transformations of an algebraically closed field of characteristic 3.Without any assumption on Morley rank, a sharply 2-transitive permutation group of characteristic 0 splits if its point stabilizers are virtually abelian.