Stable planes admitting a classical motion group
The author singles out certain automorphism groups of the classical projective planes \(P_ 2{\mathbb{F}}\) over the four real alternative division algebras \({\mathbb{R}},{\mathbb{C}},{\mathbb{H}}\) (quaternions), \({\mathbb{O}}\) (octonions) as classical motion groups (they appear as groups of automorphisms of quite natural open subsets of \(P_ 2{\mathbb{F}})\). A invariant open set of points of \(P_ 2{\mathbb{F}}\) or of the dual plane with the induced structure will be called a classical plane geometry; a detailed description of the possibilities has been given by the author in Monatsh. Math. 91, 19-37 (1981; Zbl 0453.51008). The author shows that a locally compact stable plane of dimension \(2n>0\) which admits a classical motion group of the classical 2n-dimensional projective plane \(P_ 2{\mathbb{F}}\) as a group of automorphisms is isomorphic to a classical plane geometry of the corresponding type. This gives an interesting characterization of classical plane geometries by groups, which extends results for 2-dimensional stable planes and for projective planes. For \(n=2\) this theorem fits into the author's program of determing all 4-dimensional stable planes which admit a semi-simple group of automorphisms. The proof of the theorem is devided into the 2-dimensional case \((n=1)\) and a higher-dimensional case \((n>1)\). For \(n=\dim {\mathbb{F}}>1\) all motion groups contain a subgroup \(\Phi \cong Spin_{n+1}\). Examing the possible action of \(\Phi\) the existence of isotropic elements, i.e. points for which the stabilizer acts transitively on the set of lines through this point, or dually for lines, is proved. From a previous result of the author [Math. Z. 182, 49-61 (1983; Zbl 0506.51012)] about stable planes with at least two isotropic points it follows that the considered stable plane is isomorphic to a classical subplane of \(P_ 2{\mathbb{F}}\). A comparison of the action of the entire motion group with the classical situation ends the proof.
automorphism groups of the classical projective planes
characterization of classical plane geometries by groups
scientific article; zbMATH DE number 3944689
scientific article