A stabilizer lemma for translation generalized quadrangles

AbstractLet S be a translation generalized quadrangle (TGQ) of order (s,s2), s>1 and s odd, with a good line L. Then there are precisely s3+s2 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.In this 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.Other applications will be obtained