Proceedings of the American Mathematical Society

Author(s): Koen Thas
Journal: Proc. Amer. Math. Soc. 136 (2008), 1409-1418.
MSC (2000): Primary 51E12, 20B25, 20E42
Posted: December 4, 2007
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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

-groups for some prime

1.: P. FONG AND G. M. SEITZ. Groups with a $(B,\,N)$ -pair of rank . I, II, Invent. Math. 21 (1973), 1-57; ibid. 24 (1974), 191-239. MR 0354858 (50:7335)
2.: D. FROHARDT. Groups which produce generalized quadrangles, J. Combin. Theory Ser. A 48 (1988), 139-145. MR 938864 (89d:51015)
3.: D. GORENSTEIN. Finite Groups, Second edition, Chelsea Publishing Comp., New York, 1980. MR 569209 (81b:20002)
4.: D. HACHENBERGER. Groups admitting a Kantor family and a factorized normal subgroup, Des. Codes Cryptogr. 8 (1996), 135-143. MR 1393979 (98c:20042)
5.: D. HUGHES AND F. C. PIPER. Projective Planes, Graduate Texts in Mathematics 6, Springer-Verlag, New York|Berlin, 1973. MR 0333959 (48:12278)
6.: W. M. KANTOR. Automorphism groups of some generalized quadrangles, Advances in Finite Geometries and Designs (Chelwood Gate, 1990), 251-256, Oxford Sci. Publ., Oxford Univ. Press, New York, 1991. MR 1138747 (92m:51009)
7.: N. KNARR. Private Communication, October 2004 and June 2005.
8.: S. E. PAYNE AND J. A. THAS. Finite Generalized Quadrangles, Research Notes in Mathematics 110, Pitman Advanced Publishing Program, Boston/London/Melbourne, 1984. MR 767454 (86a:51029)
9.: S. E. PAYNE AND K. THAS. Notes on elation generalized quadrangles, European J. Combin. 24 (2003), 969-981. MR 2024555 (2004h:51003)
10.: K. THAS. Automorphisms and Characterizations of Finite Generalized Quadrangles, in: Generalized Polygons, Proceedings of the Academy Contact Forum ``Generalized Polygons'' 20 October, Palace of the Academies, Brussels, Belgium (2001), 111-172.
11.: K. THAS. A theorem concerning nets arising from generalized quadrangles with a regular point, Des. Codes Cryptogr. 25 (2002), 247-253. MR 1900771 (2003e:51008)
12.: K. THAS. Automorphisms and Combinatorics of Finite Generalized Quadrangles, Ph.D. Thesis, Ghent University, Ghent (2002), xxviii+412 pp.
13.: K. THAS. Symmetry in generalized quadrangles, Des. Codes Cryptogr. 29 (2003), Proceedings of the Oberwolfach conference ``Finite Geometries (2001)'', 227-245. MR 1993170 (2004g:51003)
14.: K. THAS. Symmetry in Finite Generalized Quadrangles, Frontiers in Mathematics 1, Birkhäuser-Verlag, Basel|Boston|Berlin, 2004. MR 2036432 (2005e:51006)
15.: K. THAS. Some basic conjectures on elation generalized quadrangles, and their solutions, Bull. Belgian Math. Soc. | Simon Stevin 12 (2005), 909-918. MR 2241353 (2007d:51009)
16.: K. THAS AND S. E. PAYNE. Foundations of elation generalized quadrangles, European J. Combin. 27 (2006), 51-62. MR 2186415 (2006i:51008)
17.: K. THAS AND H. VAN MALDEGHEM. Moufang-like conditions for generalized quadrangles and classification of all quasi-transitive generalized quadrangles, Discrete Math. 294 (2005), 203-217. MR 2139798 (2006b:51005)
18.: K. THAS AND H. VAN MALDEGHEM. Geometric characterizations of finite Chevalley groups of Type $\mathbf{B}_2$ , Trans. Amer. Math. Soc., To appear.
19.: J. TITS. Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics 386, Springer-Verlag, Berlin|New York, 1974. MR 0470099 (57:9866)
20.: J. TITS AND R. M. WEISS. Moufang Polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1938841 (2003m:51008)
21.: H. VAN MALDEGHEM. Generalized Polygons, Monographs in Mathematics 93, Birkhäuser-Verlag, Basel, 1998. MR 1725957 (2000k:51004)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 51E12, 20B25, 20E42

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: 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
Posted: December 4, 2007
Additional Notes: The author is a Postdoctoral Fellow of the Fund for Scientific Research --- Flanders (Belgium).
Communicated by: Jonathan I. Hall
Copyright of article: Copyright 2007, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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