4-dimensionale projektive Ebenen vom Lenz-Barlotti-Typ II.2. (Four- dimensional projective planes of Lenz-Barlotti type II.2.)
A topological projective plane is called flexible if its collineation group has an open orbit in the flag space. A projective plane \(\pi\) is of Lenz-Barlotti type II.2 if it contains points u, v and lines S, W with \(v=W\wedge S\) and \(u\in W\setminus \{v\}\) such that \(\pi\) is (u,S)- and (w,W)-transitive and has no further (point, line)-transitivities. The author now determines all flexible 4-dimensional projective planes of Lenz-Barlotti type II.2. Every projective plane of this Lenz-Barlotti type is coordinatizable by a planar double group, and in the case treated here its additive and multiplicative groups are isomorphic to those of the complex field, respectively. Alternatively, these planes are shown to be characterized as those flexible 4-dimensional projective planes whose collineation group contains a subgroup isomorphic to \({\mathbb{R}}^ 3\times SO_ 2\).
flexible 4-dimensional projective planes of Lenz-Barlotti type
scientific article; zbMATH DE number 4017751
scientific article