Every Generalized Quadrangle of Order 5 Having a Regular Point Is Symplectic

Abstract

For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of . During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof.

2000 AMS Subject Classification::

• 05B25
• 51E12

Keywords:

• generalized quadrangle
• regular point
• (generalized) admissible triple
• affine plane

Notes

¹The code for our implementation is available at http://cage.ugent.be/~bdb.

²Code available at http://cage.ugent.be/~bdb.