On Finite Elation Generalized Quadrangles with Symmetries

We study the structure of finite groups G which act as elationgroups on finite generalized quadrangles and contain a fullgroup of symmetries about some line through the base point.Such groups are related to the translation groups of translationtransversal designs with parameters depending on those of thequadrangles. Using results on the structure of p-groups which act as translationgroups on transversal designs and results on the index of theHughes subgroups of finite p-groups, we can show how restrictedthe structure of elation groups of finite generalized quadrangleswith symmetries is. One of our main results is that G is necessarily an elementaryabelian 2-group, provided that G has even cardinality. In particular,the elation generalized quadrangle coordinatized by G is a translationgeneralized quadrangle with G as translation group, that is,G contains full groups of symmetries about every line throughthe base point.