Subquadrangles of order s of generalized quadrangles of order (s,s2), Part II

Abstract

In this paper, subquadrangles of order s of generalized quadrangles (GQ) of order ðs; s 2 Þ; with s odd, are investigated. The even case was considered in Part I. In the case where O is an egg good at an element p and the translation generalized quadrangle S ¼ TðOÞ has order ðs; s 2 Þ; with s odd, we prove that if S 0 is a subquadrangle of order s of S; then S 0 is the classical GQ Qð4; sÞ and either S is the classical GQ Qð5; sÞ or S 0 is one of the s 3 þ s 2 subquadrangles of order s containing the line p of S: Further, some characterizations of particular eggs are obtained. Finally, it is shown that if S is a flock GQ of order ðs 2 ; sÞ; s odd, with base point ðNÞ and S 0 is a subquadrangle of order s of S containing the point ðNÞ; then S is a Kantor semifield flock GQ, S 0 is isomorphic to the classical GQ W ðsÞ and either S is isomorphic to the classical GQ Hð3; s 2 Þ or S 0 is one of the s 3 þ s 2 subquadrangles of order s containing the point ðNÞ: As an application there is a characterization of the Kantor semifield flock GQ in terms of the net defined by the regular point ðNÞ of the flock GQ.

Subquadrangles of order s of generalized quadrangles of order (s,s2), Part I

“…We now prove that when s is even that given the hypotheses of Lemma 5.1 the GQ S must be the classical GQ Qð5; sÞ: Note that when s is odd the equivalent result is Theorem 4.3 of [6].…”

“…Theorem 5 of [25]. & Note that this result is valid for both s odd and even, however in the odd case Theorem 4.1 of [6] shows that such a TGQ is classical.…”

“…However, given that the characterizations of an elliptic quadric ovoid of Qð4; sÞ are much weaker than in the s even case, any such proof will be more complicated and probably longer than that given in Theorem 4.3 of [6].…”

“…Subquadrangles of order s of a TGQ with egg good at an element, containing ðNÞ Suppose that S ðNÞ ¼ ðP; B; IÞ is a TGQ of order ðs; s 2 Þ; s even, with S ¼ TðEÞ and E is good at the element d: Since E is good at d it follows that S has s 3 þ s 2 subquadrangles of order s containing both the base point ðNÞ and the line d: By Lemma 2.4 S has no other subquadrangles containing d: By Theorem 2.6 the dual net N Ã d is isomorphic to H 3 s : So if S 0 is a subquadrangle of S of order s containing ðNÞ; then S 0 is either one of the s 3 þ s 2 subquadrangles also containing d; or does not contain d and by Theorem 2.7 is isomorphic to T 2 ðOÞ for some oval O: The GQ T 3 ðOÞ; for O an ovoid of PGð3; sÞ; provide examples of GQ with such subquadrangles, and so consequently the classification of such GQ seems perhaps more difficult than in the odd case where there is only the classical example (see [6,Theorem 4.1]). In this case, we will prove only a result on the egg associated with such a GQ.…”

Subquadrangles of order s of generalized quadrangles of order (s,s2), Part III

“…This paper is a sequel to the series of papers "Subquadrangles of order s of generalized quadrangles of order (s, s 2 ), I" [3] and "Subquadrangles of order s of generalized quadrangles of order (s, s 2 ), II" [4], both authored by Brown and Thas. In this paper, we investigate subquadrangles of order s of flock generalized quadrangles (GQs) S of order (s 2 , s), s odd, with base point (∞), where the subquadrangle does not contain the point (∞). We prove that if the flock GQ has such a subquadrangle, then S is classical.…”