Proceedings of the London Mathematical Society Advance Access originally published online on January 25, 2008
Proceedings of the London Mathematical Society 2008 96(3):767-791; doi:10.1112/plms/pdm059
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© 2008 London Mathematical Society
Special Moufang sets, their root groups and their µ-maps
Department of Pure Mathematics and Computer Algebra
Ghent University
Krijgslaan 281 S22
9000 Gent
Belgium
Department of Mathematics
Ben-Gurion University
Beer-Sheva 84105
Israel
yoavs@math.bgu.ac.il
Faculty for Mathematics
University of Bielefeld
Postfach 100131
D-33501 Bielefeld
Germany
ktent@math.uni-bielefeld.de
Received 3 November 2006. Accepted 17 August 2007.
We prove Timmesfeld's conjecture that special abstract rank one groups are quasisimple. We give two characterizations of the root groups in special Moufang sets: a normal subgroup of the point stabilizer is a root group if it is either regular, or nilpotent and transitive. We prove that if a root group of a special Moufang set contains an involution, then it is of exponent 2. We also show that the root groups are abelian if and only if the so-called µ-maps are involutions.
2000 Mathematics Subject Classification 17C60, 20E42 (primary), 17C30 (secondary).
The first author is a Postdoctoral Fellow of the Research Foundation – Flanders (Belgium) (FWO – Vlaanderen). The second author is partially supported by BSF grant no. 2004-083. The third author is partially supported by DFG under SFB 701. Part of this work was done while the first two authors were guests of the University of Bielefeld and were also partially supported by the DFG under SFB 701.