Definable Envelopes of Nilpotent Subgroups of Groups with Chain Conditions on Centralizers

An $\mathfrak{M}_C$ group is a group in which all chains of centralizers have finite length. In this article, we show that every nilpotent subgroup of an $\mathfrak{M}_C$ group is contained in a definable subgroup which is nilpotent of the same nilpotence class. Definitions are uniform when the lengths of chains are bounded

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

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