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Geometric characterizations of finite Chevalley groups of type B$ _{2}$

Authors: Koen Thas and Hendrik Van Maldeghem
Journal: Trans. Amer. Math. Soc. 360 (2008), 2327-2357
MSC (2000): Primary 05B25, 51E12, 20B10, 20B25, 20E42
Published electronically: November 20, 2007
MathSciNet review: 2373316
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Abstract: Finite Moufang generalized quadrangles were classified in 1974 as a corollary to the classification of finite groups with a split BN-pair of rank $ 2$, by P. Fong and G. M. Seitz (1973), (1974). Later on, work of S. E. Payne and J. A. Thas culminated in an almost complete, elementary proof of that classification; see Finite Generalized Quadrangles, 1984. Using slightly more group theory, first W. M. Kantor (1991), then the first author (2001), and finally, essentially without group theory, J. A. Thas (preprint), completed this geometric approach. Recently, J. Tits and R. Weiss classified all (finite and infinite) Moufang polygons (2002), and this provides a third independent proof for the classification of finite Moufang quadrangles.

In the present paper, we start with a much weaker condition on a BN-pair of Type $ {\ensuremath{\mathbf{B}}_2}$ and show that it must correspond to a Moufang quadrangle, proving that the BN-pair arises from a finite Chevalley group of (relative) Type $ {\ensuremath{\mathbf{B}}_2}$. Our methods consist of a mixture of combinatorial, geometric and group theoretic arguments, but we do not use the classification of finite simple groups. The condition on the BN-pair translates to the generalized quadrangle as follows: for each point $ x$, the stabilizer of all lines through that point acts transitively on the points opposite $ x$.

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Additional Information

Koen Thas
Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Ghent, Belgium

Hendrik Van Maldeghem
Affiliation: Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Ghent, Belgium

Keywords: (Split) BN-pair, Chevalley group, Moufang condition, generalized quadrangle.
Received by editor(s): January 18, 2005
Received by editor(s) in revised form: December 28, 2005
Published electronically: November 20, 2007
Additional Notes: The first author is a Postdoctoral Fellow of the Fund for Scientific Research — Flanders (Belgium).
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.