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Journal of the London Mathematical Society Advance Access originally published online on January 10, 2008
Journal of the London Mathematical Society 2008 77(1):240-252; doi:10.1112/jlms/jdm106
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© 2008 London Mathematical Society

Uniqueness cases in odd-type groups of finite Morley rank

Alexandre V. Borovik

School of Mathematics
The University of Manchester
PO Box 88, Sackville Street
Manchester M60 1QD
United Kingdom
alexandre.borovik{at}umist.ac.uk

Jeffrey Burdges

Mathematisches Institut
Universität Würzburg
Am Hubland
D-97074 Würzburg
Germany

Ali Nesin

Mathematics Department
Istanbul Bilgi University
Kustepe Sisli
Istanbul
Turkey
anesin{at}bilgi.edu.tr

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. One of the major theorems in the area is Borovik's trichotomy theorem. The ‘trichotomy’ here is a case division of the generic minimal counterexamples within odd type, that is, groups with a large and divisible Sylow° 2-subgroup. The so-called ‘uniqueness case’ in the trichotomy theorem is the existence of a proper 2-generated core. It is our aim to drive the presence of a proper 2-generated core to a contradiction, and hence bind the complexity of the Sylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilber conjecture. This paper shows that the group in question is a minimal connected simple group and has a strongly embedded subgroup, a far stronger uniqueness case. As a corollary, a tame counterexample to the Cherlin–Zilber conjecture has Prüfer rank at most two.

2000 Mathematics Subject Classification 03C60, 20G99.

The first author completed his work on the paper during his visit to Institut Giscard Desargues, Université Lyon 1, in April 2003. The second author was partially supported by DFG grant Te 242/3-1. The third author was partially supported by the London Mathematical Society grant 4523.

Received December 6, 2004; revised April 4, 2007; published online January 10, 2008.


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