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Generalized Quadrangles of Order (s, s2), III

✍ Scribed by J.A. Thas

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 212 KB
Volume: 87
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1998.2959

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✦ Synopsis

Let S=(P, B, I) be a generalized quadrangle of order (q, q 2 ), q>1, and assume that S satisfies Property (G) at the flag (x, L). If q is odd then S is the dual of a flock generalized quadrangle. This solves (a stronger version of ) a ten-year-old conjecture. We emphasize that this is a powerful theorem as Property (G) is a simple combinatorial property, while a flock generalized quadrangle is concretely described using finite fields and groups. As in several previous theorems it was assumed that the dual of the generalized quadrangle arises from a flock, this can now be replaced, in the odd case, by having Property (G) at some flag. Finally we describe a pure geometrical construction of a generalized quadrangle arising from a flock; until now there was only the construction by Knarr which only worked in the odd case.

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