One-point extensions of generalized hexagons and octagons

In this note, we prove the uniqueness of the one-point extension S of a generalized hexagon of order 2 and prove the non-existence of such an extension S of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of S, the graph theoretic distance from y to z in the derived generalized hexagon Sx is the same as the distance from x to z in Sy. Mathematics Subject Classification 2000: 51E12. Key words and phrases: Split Cayley hexagon, one-point extension

• Type: Journal
• Year: 2006
• Where: ENDM
• Authors: Hans Cuypers, A. De Wispelaere, Hendrik Van Maldeghem
• Keywords: ENDM 2006, Finite Generalized Hexagon, Finite Generalized Octagons, Generalized Hexagon