A Characterization ofQ(5, q) Using One SubquadrangleQ(4,q )

✍ Scribed by Leen Brouns; Joseph A. Thas; Hendrik van Maldeghem

• Book ID: 102570923
• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 151 KB
• Volume: 23
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.2001.0554

✦ Synopsis

Let be a finite generalized quadrangle of order (q, q 2 ), and suppose that it has a subquadrangle isomorphic to Q(4, q). We show that is isomorphic to the classical generalized quadrangle Q(5, q) if at least one of the following holds: (1) all linear collineations of extend to ; (2) all subtended ovoids are classical (and we present a uniform proof independent of the characteristic). Further, for q odd, we prove that if every triad {x, y, z} of is 3-regular in and {x, y, z} ⊥⊥ ⊂ , then is classical. We also show that, if for every centric triad {x, y, z} of an ovoid O of the quadrangle ∼ = Q(4, q), q odd, all points of {x, y, z} ⊥⊥ belong to O, then O is classical.