Groups of finite Morley rank with a generically sharply multiply transitive action

TL;DR

The paper proves that groups of finite Morley rank with a highly transitive action on a connected abelian group are essentially general linear groups over algebraically closed fields of characteristic not 2.

Contribution

It establishes a classification of such groups, showing they are isomorphic to general linear groups over algebraically closed fields, extending understanding of group actions in model theory.

Findings

V has a vector space structure over an algebraically closed field

G acts as the general linear group on V

V has Morley rank n and no involutions

Abstract

We prove that if $G$ is a group of finite Morley rank which acts definably and generically sharply $n$-transitively on a connected abelian group $V$ of Morley rank $n$ with no involutions, then there is an algebraically closed field $F$ of characteristic $\ne 2$ such that $V$ has a structure of a vector space of dimension $n$ over $F$ and $G$ acts on $V$ as the group $\operatorname{GL}_n(F)$ in its natural action on $F^n$. This is the final pre-publication version of the paper: A. Berkman and A. Borovik, Groups of finite Morley rank with a generically sharply multiply transitive action, J. Algebra (2018), https://doi.org/10.1016/j.jalgebra.2018.07.033. Accepted for publication 28 July 2018. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published