An elementary proof of the fundamental theorem of projective geometry
The author proves the following generalization of the classical result of projective geometry which states that every isomorphism between Desarguesian projective spaces is induced by some semilinear map between the associated vector spaces. Theorem: Let \(V\), \(W\) denote vector spaces over division rings \(K\) and \(L\), respectively. Let \(g:{\mathcal P}(V) \to {\mathcal P}(W)\) be a morphism between the associated projective spaces. If the image of \(g\) is not contained in some line of \({\mathcal P}(W)\), then there exists a semilinear map \(f:V\to W\) such that \(g\) is induced by \(f\), i.e. \(g(\langle x\rangle) = \langle f(x)\rangle\) for any admissible \(x\). Moreover, as in the classical case, the map \(f\) is unique up to scalar multiplication. As an application, the author gives a short proof of Wigner's theorem.
scientific article; zbMATH DE number 1747827