A Characterisation of the Generalized Quadrangle <Emphasis Type="Italic">Q</Emphasis> (5, <Emphasis Type="Italic">q</Emphasis>) Using Cohomology

If a GQ S ′ of order (s, s) is contained in a GQ S of order (s, s2) as a subquadrangle, then for each point X of S\S ′ the set of points OX of S ′ collinear with X form an ovoid of S ′. Thas and Payne proved that if S ′ = Q(4, q), q even, and OX is an elliptic quadric for each X ∈ S\S ′, then S ∼= Q(5, q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.