Generalized quadrangles are incidence structures introduced by J. Tits in his celebrated paper [1], as a means to study finite simple groups. Finite generalized quadrangles (GQs) embedded in a finite projective space are called finite classical generalized quadrangles, and can be studied using tools from classical finite projective geometry. But GQs are also studied very intensively in a synthetic way [2], where geometrical, combinatorial, and group theoretical arguments are combined to obtain results. The study of GQs is a very active field of research in finite incidence geometry. Furthermore, GQs are used nowadays to construct codes.
Obtaining characterizations of GQs imposing group theoretic conditions is almost a field of research by itself. First of all, in this paper, the author considers only translation GQs, that is, GQs satisfying many symmetry conditions that are expressed using group theoretical conditions. The author considers a translation GQ S of order (s,s2), s > 1, s odd with a good line L. There are precisely s3+s2 subquadrangles of order s containing L, and one can consider the action of the symmetry group of S on these subquadrangles.
The author proves, as a corollary of a slightly stronger lemma, that, provided this action is transitive, the original GQ S is a classical GQ Q(5,s), that is, the GQ arising by considering the points and lines of an elliptic quadric in the finite projective space PG(5,s). This is a nice result that contributes to the general synthetic study of group theoretic characterizations of GQs.
