Genericity, Generosity, and Tori

Bruno Poizat is associated, among other things, with broadening the scope of the theory of groups of finite Morley rank to the level of stable groups [P87], in keeping with the general trend of the development of model theory from Morley to Shelah. But this has not prevented him from taking an active interest in such aspects of the theory as are associated more particularly with the case of finite Morley rank, and in particular the algebraicity conjecture of Zilber and the present author: A simple group of finite Morley rank is algebraic. In particular he joined with Borovik in one of the early manifestations of the project of bringing the techniques of finite group theory to bear on this problem, in an article on Sylow theory entitled Tores et pgroupes [BP90], two topics which, for p = 2, will be central to our discussion on this occasion. The 2-Sylow theory became the basis of a vigorous programme of close analysis, reminiscent of substantial portions of the much larger and more elaborate developments leading to the classification of the finite simple groups, and over time has achieved results considerably exceeding our initial hopes for these methods, in part because of advances on the model theoretic side, notably Wagner’s work on fields of finite Morley rank [W01]. As a result, enough is known about simple groups of finite Morley rank to give useful information about connected groups of finite Morley rank generally, particularly those containing involutions. We will illustrate this point by discussing two problems of a very general character where the structure theory can be usefully applied even though the Algebraicity Conjecture remains unresolved. The first of these was proposed by Poizat, as an example of a problem which, while trivial in an algebraic group, seemed out of reach of the theory of groups of finite Morley rank. Namely, suppose a connected group of finite Morley rank is generically of finite exponent, in other words the equation