<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
Local Sharply Transitive Actions on Finite Generalized Quadrangles
Theo Grundhöfer1 and Hendrik Van Maldeghem2
1Institut f¨ur Mathematik, Universit¨at W¨urzburg, Am Hubland, D-97074 Würzburg, Germany
2Department for Pure Mathematics and Computer Algebra, Ghent University, Krijgslaan 281,
S22, B-9000 Gent, Belgium
Annals of Combinatorics 15 (1) pp.69-80 January, 2011
AMS Subject Classification: 51E12
We classify the finite generalized quadrangles containing a line L such that some group of collineations acts sharply transitively on the ordered pentagons which start with two points of L. This can be seen as a generalization of a result of Thas and the second author [22] classifying all finite generalized quadrangles admitting a collineation group that acts transitively on all ordered pentagons, although the restriction to sharp transitivity is essential in our arguments. However, the conclusion is exactly the same family of classical generalized quadrangles (the orthogonal quadrangles and their duals). Our main result thus provides a local group theoretic characterization of these classical quadrangles.
Keywords: Moufang panel, root elations, classical generalized quadrangles


1. Barlotti, A.: Some topics in finite geometrical structures. University of North Carolina, Carolina (1965)

2. Berkovic, J.G.: Generalization of the theorems of Carter andWielandt. Soviet Math. Dokl. 7, 1525--–1529 (1966)

3. Finkel, D.,Ward, M.B.: Products of supersolvable and nilpotent finite groups. Arch. Math. (Basel) 36, 385–--393 (1981)

4. Grundhöfer, T.: Projective planes with collineation groups sharply transitive on quadrangles. Arch. Math. 43, 572--–573 (1984)

5. Grundhöfer, T., Van Maldeghem, H.: Sharp homogeneity in some generalized polygons. Arch. Math. 81, 491–--496 (2003)

6. Grundhöfer, T., Van Maldeghem, H.: Sharp homogeneity in affine planes, and in some affine generalized polygons. Abh. Mathem. Sem. Univ. Hamburg 74, 163--–174 (2004)

7. Hering, C.: On the classification of finite near fields. J. Algebra 234(2), 664–--667 (2000)

8. Holt, D.F.: Triply-transitive permutation groups in which an involution central in a Sylow 2-subgroup xes a unique point. J. London Math. Soc. (2) 15(1), 55–--65 (1977)

9. Huppert, B.: Zweifach transitive, aufiösbare permutationsgruppen. Math. Z. 68, 126–--150 (1957)

10. Huppert, B.: Endliche Gruppen. I. Springer-Verlag, Berlin-New York (1967)

11. Huppert, B., Blackburn, N.: Finite Groups. III. Springer-Verlag, Berlin-New York (1982)

12. Kantor,W.M.: Note on span-symmetric generalized quadrangles. Adv. Geom. 2, 197–--200 (2002)

13. Hering, C., Kantor, W.M., Seitz, G.M.: Finite groups with a split BN-pair of rank 1, I. J. Algebra 20, 435--–475 (1972)

14. L¨uneburg, H.: Translation Planes. Springer-Verlag, Berlin-New York (1980)

15. Payne, S.E., Thas, J.A.: Finite Generalized Quadrangles. Pitman (Advanced Publishing Program), Boston, MA (1984)

16. Passman, D.S.: Permutation Groups. W. A. Benjamin, Inc., New York-Amsterdam (1968)

17. Segre, B.: Sulle ovali nei piani lineari finiti. Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (8) 17, 141--–142 (1954)

18. Shult, E.: On a class of doubly transitive groups. Illinois J. Math. 16, 434–455 (1972)

19. Storme, L., Van Maldeghem, H.: Primitive arcs in PG(2; q). J. Combin. Theory Ser. A 69(2), 200--–216 (1995)

20. Suzuki, M.: Group Theory I. Springer-Verlag, Berlin-New York (1982)

21. Thas, J.A., Thas, K., Van Maldeghem, H.: Translation Generalized Quadrangles. World Scientific, Hackensack, NJ (2006)

22. Thas, J.A., Van Maldeghem, H.: The classification of nite generalized quadrangles admitting a group acting transitively on ordered pentagons. J. London Math. Soc. (2) 51(2), 209--–218 (1995)

23. Thas, K.: Classification of span-symmetric generalized quadrangles of order s. Adv. Geom. 2(2), 189–--196 (2002)

24. Thas, K.: A stabilizer lemma for translation generalized quadrangles. European J. Combin. 28(1), 1--–16 (2007)

25. Van Maldeghem, H.: Regular actions on generalized polygons. Int. Math. J. 2(2), 101--–118 (2002)

26. Van Maldeghem, H.: Generalized Polygons. Birkh¨auser Verlag, Basel (1998)