On finite elation generalized quadrangles with symmetries

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