Automorphism Groups of Generalized Quadrangles via an Unusual Action of PΓL (2, 2Sph)

We introduce a new, simpler method for the calculation of automorphism groups of flock quadrangles in characteristic 2, and illustrate the power of this new perspective by obtaining new characterizations of three of the four known families of flock quadrangles. Underlying this perspective is an unusual action ofPΓL(2, 2Sph) which leads to new insights about the groups of ovals in PG(2, 2Sph), and is of considerable interest independent of the connection with flock quadrangles, which arises via the herd model.