A stabilizer lemma for translation generalized quadrangles

Let S be a translation generalized quadrangle (TGQ) of order ( s , s 2 ) , s > 1 and s odd, with a good line L . Then there are precisely s 3 + s 2 subquadrangles of order s containing L . When S is isomorphic to the classical generalized quadrangle Q ( 5 , s ) , that is, the generalized quadrangle arising from a nonsingular quadric of Witt index 2 in PG ( 5 , s ) , then the stabilizer of L in the automorphism group of S acts transitively on these subquadrangles. It has been an open question for some time whether this is also the case when S is non-classical. s paper, we prove that a transitive action on these subquadrangles forces S to be isomorphic to Q ( 5 , s ) . The latter theorem is a corollary of a stronger result that will be obtained, using the proof of a ‘Stabilizer Lemma’, which allows us to interpret collineations of a semifield flock TGQ (in odd characteristic) in the associated good TGQ. applications will be obtained.