No sporadic almost simple group acts primitively on the points of a generalised quadrangle

A generalised quadrangle is a point–line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given line L point p not incident with L, there unique collinear p. They are specific case of the polygons introduced by Tits (1959), these structures their automorphism groups some importance in finite geometry. An integral part understanding quadrangles knowing which can act primitively points, particular, almost simple arise as groups. We show that no sporadic group (thick) quadrangle. also present new ideas contributing towards analysing point-primitive acting quadrangles. The first outline implementation an algorithm for determining whether second discussion conjecture resulting from observations made course this work: must either transitively lines or have exactly line-orbits, each containing half lines.