An affine proof of uniqueness for the smallest generalized quadrangles, including the determination of their automorphism groups

Abstract

We show, by elementary means, that any two generalized quadrangles with three points per line and three lines per point are isomorphic. The proof uses affine quadrangles, it also yields that the doily is point-homogeneous, and that its full group of automorphisms is isomorphic to the group of all permutations of a set of order 6. Extending the methods, we treat the case of generalized quadrangles with three points per line (and arbitrary line size) as well.