Cosets and genericity

In a connected group of finite Morley rank in which the generic element belongs to a connected nilpotent subgroup, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite Morley rank are the simple algebraic groups over algebraically closed fields and this is a motivation, among many others, for a classification project of these groups. It borrows ideas and technics from the Classification of the Finite Simple Groups but at the same time, and it is a highly nontrivial task, it provides a kind of simplified version of the finite case. This is mostly due to the existence of well-behaved notions of genericity and connectivity in the infinite case which unfortunately can hardly find finite analogs. The present short technical note deals with a very specific topic concerning such arguments based on genericity in the case of infinite groups of finite Morley rank, showing at least similarities with algebraic groups in any case as far as a theory of Weyl groups is concerned. This applies also to certain non-algebraic configurations of groups of finite Morley rank, which may not be entirely unexpected. In a simple algebraic group, maximal (algebraic) tori are conjugate and cover the group generically, with the Weyl group governing essentially the structure of the entire group. There are at least two abstract versions of tori in groups of finite Morley rank and which coincide at least in the case of a simple algebraic group, decent tori on the one hand and Carter subgroups on the other. The main caveat with these two more abstract notions is in both cases an unknown existence, more precisely the existence of a nontrivial decent torus on the one hand and the existence of a generous Carter subgroup on the other. In general “generous” for a subset of a group means “whose union of conjugates is generic in the group”, the typical property of tori in algebraic groups. With both notions there are conjugacy theorems in case of existence, or more precisely there are conjugacy theorems with nontrivial contents in case of existence. This gives a natural notion of Weyl group in each case, N(T )/C(T ) for some maximal decent torus T or N(Q)/Q for some generous Carter subgroup Q. In any case and whatever the Weyl group is, it is finite and, as with classical Weyl groups