Groups of finite Morley rank with a generically multiply transitive action on an abelian group

TL;DR

This paper characterizes highly transitive actions of finite Morley rank groups on abelian groups, showing they are essentially the natural linear actions of general linear groups over algebraically closed fields.

Contribution

It proves that such actions are equivalent to the natural linear actions of n(F) on F^n, strengthening previous results and addressing open problems.

Findings

m=n for the action's transitivity level

The action is equivalent to n(F) acting on F^n

The result applies to groups of finite Morley rank with specific transitivity properties

Abstract

We investigate the configuration where a group of finite Morley rank acts definably and generically $m$-transitively on an elementary abelian $p$-group of Morley rank $n$, where $p$ is an odd prime, and $m\geqslant n$. We conclude that $m=n$, and the action is equivalent to the natural action of $\operatorname{GL}_n(F)$ on $F^n$ for some algebraically closed field $F$. This strengthens our earlier result in arXiv:1802.05222, and partially answers two problems posed in [9].