Characterization of 16-dimensional Hughes planes

Abstract

The well-known finite Hughes planes have compact analoga with 16-dimensional point space. The automorphism group of such a plane is a 36-dimensional Lie group. Theorem: Assume that the compact projective plane $\cal P $ is not isomorphic to the classical Moufang plane over the octonions. Let $\Delta $ be a closed subgroup of $\hbox {Aut} \,\cal P $. If $\dim \Delta \ge 31$ and if $\Delta $ has a normal torus subgroup, then $\cal P $ is a Hughes plane, $\Delta = \hbox {Aut} \,\cal P $, and $\Delta ^{\prime } \cong \hbox {PSL}_3 \Bbb H $.