The class of groups of finite Morley rank arose naturally in model theoretic context, but algebraists may think of them as analogues of algebraic groups: their key properties may be expressed in terms of the axiomatic notion of dimension introduced in the early sixties by M. Morley in his studies of uncountably categorical theories. The class of groups of finite Morley rank includes the class of algebraic groups over algebraically closed fields as well as the class of finite groups. A study of groups of finite Morley rank started in the seventies in the works of B. Zil’ber and G. Cherlin. The main open problem in the field is the Classification Conjecture of Cherlin and Zil’ber: any infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field. Work on the Conjecture has made use of a variety of ideas from algebraic groups and finite groups. The first of the authors (who is a finite group theorist by training) suggested an approach to the problem which is to rework the classification of finite simple groups, but for groups of finite Morley rank instead. One of the main obstacles here is that it is not known whether every infinite simple group of finite Morley rank has an involution. It should be noted, however, that there is a real possibility that the Conjecture is false, at least in its full scope.
The book under review is a self-contained presentation of the current state of the structure theory of groups of finite Morley rank. The first five chapters are introductory, consisting of a review of basic group theory and model theory as well as the definition and basic properties of groups of finite Morley rank. The core of the book lies in Chapters 6-14 where the authors study nilpotent, solvable and semisimple groups as well as certain rings of finite Morley rank, develop a 2-Sylow theory for groups of finite Morley rank, investigate permutation groups, CN groups and CIT groups of finite Morley rank (the CIT groups are the groups with involutions where the centralizer of every involution is a two-group; the CN groups are the groups where every centralizer is nilpotent). They also consider various geometric structures related to groups of finite Morley rank.
There are three appendices. Appendix A contains miscellaneous results on groups and rings of finite Morley rank; Appendix C concerns some links with model theory. The important Appendix B is devoted to a discussion of possible direction of further research and contains a series of open problems connected with the Classification Conjecture.
The book is written by two of the main experts in the field and can help to introduce graduate students as well as practicing algebraists and logicians to the subject and, hopefully, to involve finite group theorists in a subject where there are many challenging problems. It will certainly become a standard reference book on groups of finite Morley rank.