Fibered incidence groups which are not kinematic

Abstract

An incidence space\((\beta ,\mathfrak{L})\) which is obtained from an affine space\((\beta _a ,\mathfrak{L}_a )\) by omitting a hyperplane is calledstripe space. If\((\beta _a ,\mathfrak{L}_a )\) is desarguesian, then\(\beta \) can be provided with a group operation “ ○ ” such that\((\beta ,\mathfrak{L}, \circ )\) becomes a kinematic space calledstripe group. It will be shown that there are stripe groups\((\beta ,\mathfrak{L}, \circ )\) where the incidence structure\(\mathfrak{L}\) can be replaced by another incidence structure ℜ such that\((\beta ,\Re , \circ )\) is afibered incidence group which is not kinematic. An application on translation planes concerning the group of affinities is also given.