On groups of finite Morley rank of even type
type: Item
instance of: scholarly article
title: On groups of finite Morley rank of even type
zbMATH Open document ID: 1028.20024
DOI: 10.1016/S0021-8693(03)00027-9
published in: Journal of Algebra
publication date: 2003-07-29T00:00:00Z
review text: The Cherlin-Zilber conjecture asserts that an uncountably categorical simple group (or, equivalently, a simple group of finite Morley rank) be an algebraic group over an algebraically closed field. Depending on whether its maximal 2-subgroups (2-Sylows) are finite/divisible/of finite exponent/neither of these, such a group is said to be of `degenerate'/`odd'/`even'/`mixed' type; the even type should correspond to an algebraic group over a field of characteristic 2, the odd type to an algebraic group over a field of characteristic \(\not=2\), and the degenerate and mixed types should not exist. An approach to this conjecture initiated by Borovik and Nesin studies a minimal counterexample, and follows the revisionist proof of the classification of finite simple groups. More precisely, a `\(K^*\)-group' is defined to be a group of finite Morley rank whose proper definable simple sections are algebraic groups. Thus a counterexample of minimal Morley rank would be a \(K^*\)-group. However, while \(K^*\)-groups of mixed type are known not to exist and the classification of \(K^*\)-groups of even type is nearing completion, the classification of \(K^*\)-groups of odd type is still incomplete, and \(K^*\)-groups of degenerate type may very possibly exist, thus rendering the induction impossible. The paper under review presents an inductive framework suitable for a resolution of the even type problem, first developed in the first author's habilitation thesis. Define an `\(L^*\)-group' to be a group of finite Morley rank whose proper definable simple sections are either algebraic groups over algebraically closed fields, or of odd or degenerate type. Then all simple groups of finite Morley rank of even type are algebraic if and only if there is no simple non-algebraic \(L^*\)-group of even type. The authors express hope that this can be shown by generalizing the analogous result for \(K^*\)-groups. As a first indication they show Theorem 1. If \(M\) is a weakly embedded subgro…
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 1955952
zbMATH Keywords: simple groups
zbMATH Keywords: groups of finite Morley rank
zbMATH Keywords: \(K^*\)-groups
zbMATH Keywords: \(L^*\)-groups
zbMATH Keywords: weakly embedded subgroups
zbMATH Keywords: even type
zbMATH Keywords: mixed type
zbMATH Keywords: degenerate type
author: Tuna Altınel
author: Gregory Cherlin
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1955952