The focus of this post is elation generalised quadrangles. These are generalised quadrangles defined with respect to certain automorphisms of the generalised quadrangle. My main sources for this post have been Michel (Celle) Lavrauw’s and Maska Law’s Phd theses which are both availiable from Ghent’s PhD theses in finite geometry page.

Let ${\mathcal{Q}}$ be a generalised quadrangle of order ${(s,t)}$ with a point ${x}$. An elation about ${x}$ is an automorphism of ${\mathcal{Q}}$ that fixes ${x}$, fixes each line incident with ${x}$ and fixes no point not collinear with ${x}$. If there exists a group ${G}$ of order ${s^2t}$ consisting entirely of elations and which acts regularly on the set of points not collinear with ${x}$ then ${\mathcal{Q}}$ is called an elation generalised quadrangle or simply an EGQ. The group ${G}$ is called an elation group and ${x}$ is called the base point.