Generalized Quadrangles of Order (s, s2), III

Let S=(P, B, I) be a generalized quadrangle of order (q, q2), 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.