Two remarks on generalized hexagons
The only known finite generalized hexagons were presented by \textit{J. Tits} in his famous triality paper [Publ. Math. Inst. Hautes Étud. Sci. 2, 13--60 (1959; Zbl 0088.37204)] and belong to two infinite families associated with finite simple groups. These are the only finite Moufang hexagons (i.e., admitting collineation groups with certain specified transitivity properties). The hexagons related to the groups \(G_2(q)\) are called the split Cayley hexagons and denoted \(H(q)\). In the paper under review the authors give some technical but quite interesting combinatorial characterizations of the known hexagons, and especially of the \(H(q)\) with \(q\) even. The definitions seem a bit involved for a review, but these results build on the work of \textit{J. van Bon}, \textit{H. Cuypers} and \textit{H. Van Maldeghem} [Forum Math. 8, No. 3, 343--362 (1994; Zbl 0854.51004)] and on the much earlier work of \textit{M. A. Ronan} [Geom. Dedicata 11, 61--67 (1981; Zbl 0485.51014)].
scientific article; zbMATH DE number 1443428