Elation generalized quadrangles with extra automorphisms and trivial spans

We show that the parameters of a finite elation generalized quadrangle for which the point spans containing the elation point (infinity) are trivial, and which admit extra automorphisms fixing (infinity) linewise, are powers of the same prime, so that the elation group is a p-group for some prime p. This observation has several strong corollaries, perhaps the most remarkable of which is that a thick elation quadrangle of order (s, s(2)) (for the natural number s) that admits an extra automorphism of the aforementioned type which is an involution, is a translation generalized quadrangle-that is, the elation group is elementary abelian. (c) 2008 Published by Elsevier Ltd.