Groups of Lie type acting on generalised quadrangles

Jacques Tits introduced generalised polygons to realise various groups of Lie type as symmetries of geometric objects. Since then, mathematicians were interested in studying and classifying these objects. It is known that if a group $G$ acts point and line-primitively on a generalised quadrangle, then $G$ must be almost simple. Moreover, if $G$ is also flag-transitive, then $G$ must be almost simple of Lie type. In this thesis, we show that an almost simple group $G$ acting point and line-primitively on a generalised quadrangle cannot have $\soc{G}$ isomorphic to $\Sz{2^{2m+1}}$, $\Ree{3^{2m+1}}$ or$\LargeRee{2^{2m+1}}$ where $m$ is a positive integer.