Some basic questions and conjectures on elation generalized quadrangles, and their solutions

In this paper, I will start with posing three fundamental and old questions on (elation) generalized quadrangles, and survey tersely answers on these questions coming from recent work of S. E. Payne and the author of this paper. I will then introduce a fourth question posed recently by S. E. Payne, and will provide a general answer to this question, a result independently obtained by R. Rostermundt for the Hermitian quadrangles H (3, q(2)), q even, in an entirely different fashion. Finally, I will show that this answer yields examples of elation generalized quadrangles for which the automorphism group fixing the elation point is not induced by the automorphisms of the elation group fixing the associated 4-gonal family.