Primitive permutation groups of finite Morley rank and affine type

TL;DR

This paper reviews the development of the theory of primitive permutation groups of finite Morley rank, focusing on affine type groups with abelian normal subgroups, highlighting their significance in model theory and group actions.

Contribution

It provides a comprehensive overview of primitive groups of affine type within the context of finite Morley rank, emphasizing their structural properties and role in model-theoretic group analysis.

Findings

Primitive groups of affine type are central in finite Morley rank theory.

Connected groups with abelian normal subgroups act primitively on sets.

The study clarifies the role of these groups as analogues of Galois groups.

Abstract

We give a review of one of the lines in development of the theory of groups of finite Morley rank. These groups naturally appear in model theory as model-theoretic analogues of Galois groups, therefore their actions and their role as permutation groups is of primary interest. We restrict our story to the study of connected groups of finite Morley rank $G$ acting in a definably primitive way on a set $X$ and containing a definable abelian normal subgroup $V$ which acts on $X$ regularly -- the so-called \emph{primitive groups of affine type}. For reasons explained in the paper, this case plays a central role in the theory.