Actions of $\operatorname{Alt}(n)$ on groups of finite Morley rank without involutions

TL;DR

This paper studies how the alternating group (n) acts on connected groups of finite Morley rank without involutions, establishing bounds on the rank and implications for permutation group transitivity.

Contribution

It provides new lower bounds on the rank of groups of finite Morley rank admitting (n) actions without involutions, extending previous results to the nonsolvable case.

Findings

Established a lower bound of n on the rank of such groups

Proved bounds for solvable groups using recent abelian case results

Applied findings to limits on generic transitivity in permutation groups

Abstract

We investigate faithful representations of $\operatorname{Alt}(n)$ as automorphisms of a connected group $G$ of finite Morley rank. We target a lower bound of $n$ on the rank of such a nonsolvable $G$, and our main result achieves this in the case when $G$ is without involutions. In the course of our analysis, we also prove a corresponding bound for solvable $G$ by leveraging recent results on the abelian case. We conclude with an application towards establishing natural limits to the degree of generic transitivity for permutation groups of finite Morley rank.