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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Solution of a question of Knarr


Author: Koen Thas
Journal: Proc. Amer. Math. Soc. 136 (2008), 1409-1418
MSC (2000): Primary 51E12, 20B25, 20E42
Published electronically: December 4, 2007
MathSciNet review: 2367114
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Abstract | References | Similar Articles | Additional Information

Abstract: The Moufang condition is one of the central group theoretical conditions in Incidence Geometry, and was introduced by Jacques Tits in his famous lecture notes (1974).

About ten years ago, Norbert Knarr studied generalized quadrangles (buildings of Type $ \mathbf{B}_2$) which satisfy one of the Moufang conditions locally at one point. He then posed the fundamental question whether the group generated by the root-elations with its root containing that point is always a sharply transitive group on the points opposite this point, that is, whether this group is an elation group.

In this paper, we solve the question and a more general version affirmatively for finite generalized quadrangles.

Moreover, we show that this group is necessarily nilpotent (which was only known up till now when both Moufang conditions are satisfied for all points and lines).

In fact, as a corollary, we will prove that these groups always have to be $ p$-groups for some prime $ p$.


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Additional Information

Koen Thas
Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281, S22, B-9000 Ghent, Belgium
Email: kthas@cage.UGent.be

DOI: http://dx.doi.org/10.1090/S0002-9939-07-09136-8
PII: S 0002-9939(07)09136-8
Received by editor(s): June 10, 2005
Received by editor(s) in revised form: October 31, 2006, and November 23, 2006
Published electronically: December 4, 2007
Additional Notes: The author is a Postdoctoral Fellow of the Fund for Scientific Research — Flanders (Belgium).
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.