Smooth Hughes planes are classical

Abstract

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers $\Bbb C $ , the quaternions $\Bbb H $ , and the Caley numbers $\Bbb O $ . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension $d \geq 17$ is isomorphic to the quaternion projective plane ${\cal P _2\Bbb H }$ . For topological compact projective planes this is true if $d \geq 19$ , and this bound is sharp.