Foundations of elation generalized quadrangles

In these notes we are interested in the following fundamental question: “Given a thick generalized quadrangle  S(x)S(x)with elation point (respectively center of transitivity)  xx, when does the set of all elations about  xxform a group?”. It was the general belief for a long time that this is usually the case. However, the goal of these notes is to show that this belief is wrong. To that end, we will answer the question for all known (classes of) generalized quadrangles.