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Good tori in groups of finite Morley rank. (English) Zbl 1083.20026

Recall that a ‘torus’ in a group of finite Morley rank is a definable divisible Abelian subgroup; it is ‘decent’ if it is the definable hull of its torsion elements, and ‘good’ if every definable subgroup is decent. The author shows that good tori have strong rigidity properties: (1) A connected definable group of automorphisms is trivial. (2) A uniformly definable family of subgroups is finite. (3) Any uniformly definable family of homomorphisms from a group \(H\) to a good torus is finite. This is used to deduce that good, and also decent tori in a group of finite Morley rank are conjugate, and the union of the connected components of their centralizers is generic.
The main idea is the nongenericity lemma on page 617, which should hopefully find more applications.
Note that the notion of good torus plays an important rôle in the classification of simple groups of finite Morley rank.

MSC:

20E34 General structure theorems for groups
20A15 Applications of logic to group theory
03C45 Classification theory, stability, and related concepts in model theory
20E32 Simple groups
20E07 Subgroup theorems; subgroup growth
20E36 Automorphisms of infinite groups
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