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Journal of the London Mathematical Society Advance Access originally published online on November 16, 2007
Journal of the London Mathematical Society 2008 77(1):1-14; doi:10.1112/jlms/jdm088
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© 2007 London Mathematical Society

A new trichotomy theorem for groups of finite Morley rank

Alexandre Borovik and Jeffrey Burdges

School of Mathematics
The University of Manchester
PO Box 88
Sackville Street
Manchester
M60 1QD
United Kingdom
Alexandre.Borovik{at}manchester.ac.uk

There is a longstanding conjecture, of Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Here we will conclude that a simple K*-group of finite {M}orley rank and odd type either has normal rank of at most 2, or else is an algebraic group over an algebraically closed field of characteristic not 2. To this end, it suffices to produce a proper 2-generated core in groups with \Prufer rank 2 and normal rank at least 3, which is what is proved here. Our final conclusion constrains the Sylow 2-subgroups available to a minimal counterexample and, finally, proves the trichotomy theorem in the nontame context.

2000 Mathematics Subject Classification 03C60 (primary), 20G99 (secondary).

The second author was supported by NSF grant DMS-0100794 and Deutsche Forschungsgemeinschaft grant Te 242/3-1.

Received October 1, 2005; revised March 6, 2007; published online November 16, 2007.


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