, Volume 73, Issue 1, pp 73-80

Smooth Hughes planes are classical

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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.

Received: 30.3.1998