A characterization of the finite Moufang hexagons by generalized homologies (Q1183099)

Let \(\mathcal S\) be a finite thick generalized hexagon [see the paper of \textit{W. M. Kantor}, Lect. Nots Math. 1181, 79-158 (1986; Zbl 0599.51015)]. A generalized homology of \(\mathcal S\) is an automorphism of \(\mathcal S\) fixing all points on two mutually opposite lines or fixing all lines through two mutually opposite points. The author shows that if \(\mathcal S\) admits ``many'' generalized homologies, then \(\mathcal S\) is Moufang and hence \(\mathcal S\) or its dual is isomorphic to \(G_ 2(q)\) or to \(^ 3D_ 4(q)\) for some prime power \(q\). Here \(G_ 2(q)\) and \(^ 3D_ 4(q)\) denote classical finite generalized hexagons related to the Chevalley groups \(G_ 2(q)\) and \(^ 3D_ 4(q)\), respectively.