Compact 16-dimensional projective planes (Q1290402)

The author continues his study of \(16\)-dimensional compact projective planes with large groups of automorphisms. It is shown that the existence of a group \(\Delta\) of automorphisms with \(\dim\Delta=39\) such that \(\Delta\) fixes more than one point implies that the plane is a translation plane. Then one knows that the plane is coordinatized by a so-called perturbation of the octonions [see 82.4 in \textit{H. Salzmann} et al., `Compact projective planes' (1995; Zbl 0851.51003)]. The proof in the present paper uses previous results that imply that \(\Delta\) contains a normal vector group. A detailed study of the linear representation of \(\Delta\) on this vector group yields the theorem.