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Groups of mixed type. (English) Zbl 0878.20003

The paper under review deals with groups of finite Morley rank. The current state of the structure theory of groups of finite Morley rank is presented in the book by A. Borovik, A. Nesin [Groups of finite Morley rank, Clarendon Press, Oxford (1994; Zbl 0816.20001)]. The main stimulus in the field is the open Cherlin-Zil’ber Conjecture which proposes that infinite simple groups of finite Morley rank are algebraic groups over algebraically closed fields. The authors present a program of investigation of groups of finite Morley rank which gives an approach to this problem. The approach is inspired by the ideas from finite group theory developed in the classification of finite simple groups.
The main result of the paper is a proof of a special case of the following conjecture (which is a part of the program and which should be true if the Cherlin-Zil’ber Conjecture is true): a simple group of finite Morley rank cannot be of mixed type. Here a group of finite Morley rank is said to be of mixed type if it is neither of odd type nor of even type; it is called of odd type if its Sylow subgroups are of bounded exponent; it is called of even type if the connected component of any of its Sylow subgroups is divisible abelian. Specifically, the authors prove the conjecture for the so called tame \(K^*\)-groups; these groups naturally arise when one considers minimal counterexamples to certain versions of the Cherlin-Zil’ber Conjecture.

MSC:

20A15 Applications of logic to group theory
03C60 Model-theoretic algebra
20E32 Simple groups
03C45 Classification theory, stability, and related concepts in model theory
20G15 Linear algebraic groups over arbitrary fields
20D05 Finite simple groups and their classification
20E34 General structure theorems for groups

Citations:

Zbl 0816.20001
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References:

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