On the classification of 16-dimensional planes (Q1580087)

Let \(\mathcal P\) be a compact projective plane of dimension 16, and let \(\Delta\) be a closed connected subgroup of the automorphism group of \(\mathcal P\). If \(\dim\Delta\geq 27\) then \(\Delta\) is a Lie group [\textit{B. Priwitzer} and \textit{H. Salzmann}, J. Lie Theory 8, No. 1, 83-93 (1998; Zbl 0902.51012)]. In the paper under review, the author assumes \(\dim\Delta\geq 33\) and that \(\mathcal P\) is not a Hughes plane. He proves: if \(\Delta\) is not semisimple then, up to duality, there is a minimal closed connected normal subgroup \(\Theta\cong{\mathbb R}^t\) consisting of axial collineations with common axis. Either \(\Theta\) is a group of homotheties (and \(\dim\Theta=1\)), or \(\Theta\) consists of elations. It should be noted that, although one starts the proof with some minimal connected closed normal subgroup \(\Xi\), it may be necessary to construct \(\Theta\) different from \(\Xi\). The proof consists of a careful study of the linear representation of \(\Delta\) on \(\Xi\) and on \(\Theta\). The cases where \(\dim\Xi\geq 8\) is even, are quite involved. The case where \(\Delta\) is semisimple was discussed by \textit{B. Priwitzer} [Arch. Math. 68, No. 5, 430-440 (1997; Zbl 0877.51014) and Monatsh. Math. 127, No. 1, 67-82 (1999; Zbl 0929.51010)], the 16-dimensional Hughes planes were characterized by \textit{H. Salzmann} [Arch. Math. 71, No. 3, 249-256 (1998; Zbl 0926.51016)].