Generalized Quadrangles Admitting an Abelian Singer Group.
Koen Thas (Universiteit Gent)
Let S be a thick finite generalized quadrangle (GQ) of order (s,t). We say that S admits a Singer group G if G is a collineation group acting regularly on its points. In the beginning of the 90's, D. Ghinelli wrote a paper on GQs admitting this type of group, proving several kinds of existence and non-existence results. Amongst these, she showed that when s = t and G is abelian, such a GQ cannot exist.
Recently, in a joint work with S. De Winter, we have classified all thick finite GQs admitting an abelian Singer group. In this talk, I want to report on this result, and corollaries of the proof.