Permutation groups of Finite Morley rank

Introduction Groups of finite Morley rank made their first appearance in model theory as binding groups, which are the key ingredient in Zilber's ladder theorem and in Poizat's explanation of the Picard-Vessiot theory. These are not just groups, but in fact permutation groups acting on important definable sets. When they are finite, they are connected with the model theoretic notion of algebraic closure. But the more interesting ones tend to be infinite, and connected. Many problems in finite permutation group theory became tractable only after the classification of the finite simple groups. The theory of permutation groups of finite Morley rank is not very highly developed, and while we do not have anything like a full classification of the simple groups of finite Morley rank in hand, as a result of recent progress we do have some useful classification results as well as some useful structural information that can be obtained without going through an explicit classification. So it seems like a good time to review the situation in the theory of permutation groups of finite Morley rank and to lay out some natural problems and their possible connections with the body of research that has grown up around the classification effort. The study of transitive permutation groups is equivalent to the study of pairs of groups (G, H) with H a subgroup of G, and accordingly one can read much of general group theory as permutation group theory, and vice versa, and, indeed, a lot of what goes on in work on classification makes a good deal of sense as permutation group theory—including even the final identification of a group as a Chevalley group, which can go via Tits' theory of buildings, or in other words by recognition of the natural permutation representations of such groups. Many special topics in permutation groups tied up with structural issues were discussed in [7, Chapter 11], with an eye toward applications. See also Part III of [15]. Both authors thank the Newton Institute, Cambridge, for its hospitality during the Model Theory and Algebra program, where the bulk of this work was carried out, as well as CIRM for its hospitality at the September 2004 meeting on Groups, Geometry and Logic, where the seed was planted. Thanks to Altınel for continued discussions all along the way. The most important class of permutation groups consists of the definably primitive permutation groups, and in finite …