Transformations of Grassmannians preserving the class of base subsets (Q1879102)

Let \({\mathcal P}\) denote an \(n\)-dimensional projective space and let \({\mathcal G}_k\), \(1 \leq k \leq n-2\), denote the Grassmannian consisting of all \(k\)-dimensional subspaces of \({\mathcal P}\). If \(B\) is a base for \({\mathcal P}\), then the set consisting of all \(k\)-dimensional subspaces of \({\mathcal P}\) spanned by points of \(B\) will be called the base subset of \(\mathcal{G}_k\) associated with \(B\). The author proves that if \(f\) is a surjective transformation of \({\mathcal G}_k\) sending base subsets to base subsets, then one of the following cases occurs: (1) \(n \not= 2k+1\) and \(f\) is induced by a collineation of \({\mathcal P}\) to itself, (ii) \(n = 2k+1\) and \(f\) is induced by a collineation of \({\mathcal P}\) to itself or to its dual projective space \({\mathcal P}^\ast\).