Pushing up and \(C(G,T)\) in groups of finite Morley rank of even type
type: Item
instance of: scholarly article
title: Pushing up and \(C(G,T)\) in groups of finite Morley rank of even type
zbMATH Open document ID: 1001.20023
published in: Journal of Algebra
publication date: 2002-09-24T00:00:00Z
review text: The long standing Cherlin-Zil'ber Conjecture is that any infinite simple group of finite Morley rank is an algebraic group over an algebraically closed field. The paper under review is a part of a series aimed ultimately at proving the Even Type Conjecture -- a special case of the Cherlin-Zil'ber Conjecture for groups of even type without infinite definable simple sections of degenerate type. Here a group of finite Morley rank is said to be of degenerate (even) type if its Sylow 2-subgroups are finite (infinite of bounded exponent). The authors study simple \(K^*\)-groups of even type. (A group of finite Morley rank is called a \(K^*\)-group if every infinite proper definable simple section of it is algebraic over an algebraically closed field; \(K^*\)-groups naturally arise when one considers minimal counterexamples.) The results are too technical to be formulated here; they are analogs of some results from finite group theory in the context of groups of finite Morley rank (in particular, see \textit{B.~Baumann} [Proc. Am. Math. Soc. 74, 215-222 (1979; Zbl 0409.20009)] and \textit{B.~Stellmacher} [Arch. Math. 46, 8-17 (1985; Zbl 0588.20013)]). In fact, a third of the paper consists of an appendix where a proof of an analog of Stellmacher's result is presented.
reviewed by: Oleg Belegradek
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 1740516
zbMATH Keywords: groups of finite Morley rank
zbMATH Keywords: infinite simple groups
zbMATH Keywords: Cherlin-Zil'ber conjecture
zbMATH Keywords: algebraic groups
zbMATH Keywords: Sylow subgroups
author: Tuna Altınel
author: Alexandre V. Borovik
author: Gregory Cherlin
MaRDI profile type: Publication
full work available at URL: jabr.2001.8950
OpenAlex ID: W1966661756
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DOI: 10.1006/JABR.2001.8950
description: scientific article; zbMATH DE number 1740516