On Weyl groups in minimal simple groups of finite Morley rank

Abstract

Abstract. We prove that generous non-nilpotent Borel subgroups of connected minimal simple groups of finite Morley rank are self-normalizing. We use this to introduce a uniform approach to the analysis of connected minimal simple groups of finite Morley rank through a case division incorporating four mutually exclusive classes of groups. We use these to analyze Carter subgroups and Weyl groups in connected minimal simple groups of finite Morley rank. Finally, the self-normalization theorem is applied to give a new proof of an important step in the classification of simple groups of finite Morley rank of odd type. IntroductionThe theory of groups of finite Morley rank has become a well-established branch of the model theory of groups. Most research is centered around the analysis of the infinite simple ones and more precisely the following conjecture independently stated by Gregory Cherlin and Boris Zil'ber.Algebraicity Conjecture (Cherlin-Zil'ber): An infinite simple group of finite Morley rank, seen as a pure group structure, is a linear algebraic group over an algebraically closed field.The most important advances in the analysis of the infinite simple groups of finite Morley rank have drawn upon three mathematical resources: the theory of linear algebraic groups over algebraically closed fields, the classification of the finite simple groups, and the analysis of the inherent structure of groups of finite Morley rank using a kind of geometric approach that relying on genericity arguments.As in the classification of the finite simple groups, the most difficult questions, some of which are still open, have arisen in the analysis of "small" simple groups, in particular, the connected minimal simple groups. These are the ones whose proper, definable, connected subgroups are solvable. Their analysis, that widely varies from one configuration to another, nevertheless permits impressive achievements as soon as the definable subgroup structure is sufficiently rich, a phenomenon reminiscent of the Feit-Thompson theorem in finite group theory.

Structure of Borel subgroups in simple groups of finite Morley rank

“…The three main references are [Del08] that generalizes the fundamental [CJ04] to a non-tame context, [BD09] and [ABF12]. The main theme is the notion of the Weyl group of a group G of finite Morley rank, denoted W (G), and defined as N G (T )/C G (T ) where T is any maximal decent torus of G. By Fact 2.21, the Weyl group of a group of finite Morley rank G is well-defined.…”

“…In the rest of this section, we will recall various facts from [ABF12]. The following theorem, that will not be directly used in this article, is involved in the proofs of many other crucial facts such as 3.9, 3.12, 3.13.…”

“…In [ABF12], a uniform approach to the analysis of connected minimal simple groups of finite Morley rank was introduced through a case division that consists of four mutually exclusive classes of groups. This case division follows two criteria: generic disjointness of Borel subgroups from their conjugates and (non-)triviality of the Weyl group.…”

“…Indeed, the condition (*) describes a strict subclass of groups of type (1), and it in particular excludes the bad groups. We recall some remarks from [ABF12] concerning the four types of groups. • The minimal connected simple groups with a nongenerous Carter subgroup are of type (2) and are analyzed in [Fré08].…”

“…The groups not covered by our theorems form a subclass of groups of type (1), one of the four mutually exclusive families that cover all connected minimal simple groups and were introduced in [ABF12] (Fact 3.9). The groups satisfying the (*) hypothesis are groups whose structure is close to that of a bad group (see Fact 2.12), but having no nilpotent Borel subgroup.…”