Groups of finite Morley rank with solvable local subgroups. (English) Zbl 1248.20039
This paper is the first in a series of four papers on groups of finite Morley rank with soluble local subgroups [J. Reine Angew.Math.644, 23-45 (2010; Zbl 1203.20033); arXiv:0802.1394 (in preparation)] in which the authors undertake to transfer the theory of minimal connected simple groups of finite Morley rank to \(*\)-locally soluble groups of finite Morley rank. Here a group \(G\) of finite Morley rank is \(*\)-locally soluble if normalizers of non-trivial Abelian subgroups are soluble; it is \(*\)-locally\(^0\) soluble if the connected components of those normalizers are soluble, \(*\)-locally\(_0\) soluble if the normalizers of connected non-trivial Abelian subgroups are soluble, and \(*\)-locally\(^0_0\) soluble if the connected components of those normalizers are soluble.
The authors prove a uniqueness theorem analogous to the Bender method for finite group theory, and derive its corollaries.
The paper essentially reproves known results in this wider context, often referring back to the original papers. Its principal aim is to prepare the grounds for the rest of the series.
The authors prove a uniqueness theorem analogous to the Bender method for finite group theory, and derive its corollaries.
The paper essentially reproves known results in this wider context, often referring back to the original papers. Its principal aim is to prepare the grounds for the rest of the series.
Reviewer: Frank Wagner (Villeurbanne)
MSC:
| 20F11 | Groups of finite Morley rank |
| 03C60 | Model-theoretic algebra |
| 20E25 | Local properties of groups |
| 20E34 | General structure theorems for groups |
| 20F19 | Generalizations of solvable and nilpotent groups |
| 03C45 | Classification theory, stability, and related concepts in model theory |
Keywords:
Bender method; algebraicity conjecture; groups of finite Morley rank; local subgroups; soluble subgroups; groups of even type; groups of odd type; Cherlin-Zilber conjectureCitations:
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