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 ( ∞ ) are trivial, and which admit extra automorphisms fixing ( ∞ ) linewise, are powers of the same prime, so that the elation group is a p -group for some prime p . bservation 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.