Generalized quadrangles and regularity

Matthew R. Brown
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let S = (P, B, I) be a generalized quadrangle of order (s, t), s, t > 1, and assume that S has a regular point X. In this paper we survey some basic results on such generalized quadrangles (GQs) as well as the known examples. We also study a general representation of such GQs using the net associated with the regular point X and specialise the representation to the case where X is abelian centre of symmetry, as in all of the known examples. Perhaps the most important structural result
more &raquo; ... regularity in GQs is the following theorem giving a construction of a (dual) net from a regular point of a GQ. Theorem 6 (1.3.1 of Payne and Thas [26] ). Let S have a regular point X. Then the incidence structure with pointset X ⊥ \{X}, with lineset the set with elements {Y, Z} ⊥⊥ , where Y, Z ∈ X ⊥ \{X}, Y / ∼ Z, and with the natural incidence, is the dual of a net of order s and degree t + 1. If in particular s = t > 1, then there arises a dual affine plane of order s. Moreover, in this case the incidence structure X with pointset X ⊥ , with lineset the set with elements {Y, Z} ⊥⊥ , where Y, Z ∈ X ⊥ \{X}, and with the natural incidence, is a projective plane of order s.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1016/j.disc.2004.04.034</a> <a target="_blank" rel="external noopener" href="">fatcat:t6njkpxo2fdkfasifwaxradj6a</a> </span>
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