Structure of linear groups definable in a field of finite Morley rank.
type: Item
instance of: scholarly article
title: Structure of linear groups definable in a field of finite Morley rank.
zbMATH Open document ID: 1070.20056
author: Yerulan Mustafin
published in: Journal of Algebra
publication date: 2004-11-26T00:00:00Z
review text: The author uses \textit{B. Poizat}'s identification of simple linear groups of finite Morley rank in positive characteristic as algebraic groups [J. Symb. Log. 66, No. 4, 1637-1646 (2001; Zbl 1005.03039)] and an analysis of unipotent groups in characteristic zero to prove a structure theorem for a linear group \(G\) of finite Morley rank: In positive characteristic \(G/R(G)\) is isomorphic to a finite product of simple algebraic groups, in characteristic zero \(G/R(G)\) is the product of a definable group without unipotent elements and finitely many simple algebraic groups. As a consequence he obtains conjugacy of the Borel and Carter subgroups, as well as of the maximal tori. Similar results hold for linear groups of Morley rank \(\omega n\) and characteristic zero, provided no groups of finite rank are definable.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 2119121
zbMATH Keywords: simple linear groups
zbMATH Keywords: groups of finite Morley rank
zbMATH Keywords: semi-simple groups
zbMATH Keywords: unipotent groups
zbMATH Keywords: Borel subgroups
zbMATH Keywords: Carter subgroups
zbMATH Keywords: tori
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.07.017
OpenAlex ID: W2093794058
cites work: Simple \(L^*\)-groups of even type with strongly embedded subgroups.
cites work: A Hall Theorem for ω-Stable Groups
cites work: Superstable groups
cites work: Q4317873
cites work: Tame minimal simple groups of finite Morley rank
cites work: Superstable fields and groups
cites work: Abnormal subgroups in solvable groups of finite Morley rank
cites work: Q4089001
cites work: Primitive Permutation Groups of Finite Morley Rank
cites work: On $ω_1$-categorical theories of abelian groups
cites work: On $ω_1$-categorical theories of fields
cites work: Q5700781
cites work: Generalized Fitting subgroup of a group of finite Morley rank
cites work: Q3028990
cites work: Quelques modestes remarques a propos d'une conséquence inattendue d'un résultat surprenant de monsieur frank olaf wagner
cites work: L'égalité au cube
cites work: Nilpotent complements and Carter subgroups in stable \({\mathfrak R}\)
cites work: Fields of finite Morley rank
cites work: BAD FIELDS IN POSITIVE CHARACTERISTIC
DOI: 10.1016/J.JALGEBRA.2004.07.017
description: scientific article; zbMATH DE number 2119121