Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-26T01:26:12.399Z Has data issue: false hasContentIssue false
  • English
  • Français

Local identification of spherical buildings and finite simple groups of Lie type

Published online by Cambridge University Press:  07 February 2013

ULRICH MEIERFRANKENFELD ,
GERNOT STROTH  and
RICHARD M. WEISS
ULRICH MEIERFRANKENFELD
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824, U.S.A. e-mail: meier@math.msu.edu
GERNOT STROTH
Affiliation:
Institut für Mathematik, Martin–Luther–Universität Halle–Wittenberg, 06099 Halle (Saale), Germany e-mail: gernot.stroth@mathematik.uni-halle.de
RICHARD M. WEISS
Affiliation:
Department of Mathematics, Tufts University, Medford, MA 02155, U.S.A. e-mail: Richard.Weiss@tufts.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We give a short proof of the uniqueness of finite spherical buildings of rank at least 3 in terms of the structure of the rank 2 residues and use this result to prove a result making it possible to identify an arbitrary finite group of Lie type from knowledge of its “parabolic structure” alone. Our proof also involves a connection between loops, “Latin chamber systems” and buildings.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 3 , May 2013 , pp. 527 - 547
Copyright
Copyright © Cambridge Philosophical Society 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aschbacher, M.Finite geometries of type C 3 with flag transitive automorphism groups. Geom. Dedicata 16 (1984), 195–200.CrossRefGoogle Scholar
[2]Carter, R. W.Simple Groups of Lie Type. Pure and Applied Mathematics 28 (John Wiley & Sons, 1972).Google Scholar
[3]Fong, P. and Seitz, G.Groups with a (B,N)-pair of rank 2, I. Invent Math. 21 (1973), 1–57.CrossRefGoogle Scholar
[4]Meixner, T. and Timmesfeld, F.Chamber systems with string diagrams. Geom. Dedicata 15 (1983), 115–123.CrossRefGoogle Scholar
[5]Pickert, G.Projektive Ebenen (Springer-Verlag, 1955).CrossRefGoogle Scholar
[6]Schreier, O. and van der Waerden, B.Die Automorphismen der projektiven Gruppen. Abh. math. Sem. Hamburg 6 (1928), 303–322.CrossRefGoogle Scholar
[7]Tits, J.A local approach to buildings. In The Geometric Vein: The Coxeter Festschrift, Davis, C., Grünbaum, B. and Sherk, F.A., eds. (Springer-Verlag, 1981), pp. 519–547.CrossRefGoogle Scholar
[8]Tits, J.Buildings of Spherical Type and Finite BN-Pairs Lecture Notes in Math. 386 (Springer-Verlag, 1974).Google Scholar
[9]Tits, J. and Weiss, R. M.Moufang Polygons (Springer-Verlag, 2002).CrossRefGoogle Scholar
[10]Weiss, R. M.The Structure of Spherical Buildings (Princeton University Press, 2003).Google Scholar
[11]Zorn, M.Theorie der alternativen Ringe. Abh. Math. Seminar Hamburg 8 (1930), 123–147.CrossRefGoogle Scholar