A stabilizer lemma for translation generalized quadrangles

Let SS be a translation generalized quadrangle (TGQ) of order (s,s2)(s,s2), s>1s>1 and ss odd, with a good line LL. Then there are precisely s3+s2s3+s2 subquadrangles of order ss containing LL. When SS is isomorphic to the classical generalized quadrangle Q(5,s)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 LL in the automorphism group of SS acts transitively on these subquadrangles. It has been an open question for some time whether this is also the case when SS is non-classical.In this paper, we prove that a transitive action on these subquadrangles forces SS to be isomorphic to Q(5,s)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.