Crystallographic point groups of five-dimensional space. 1. Their elements and their subgroups

The purpose of this work is to introduce a method with a view to obtaining the crystallographic point groups of five-dimensional space, i.e. the subgroups of the holohedries of these space crystal families. This paper is specifically devoted to numerical analysis, whereas the following ones deal with some applications to crystallography. These results have been obtained through a collaboration between two teams: H. Veysseyre (Institut Supérieur des Matériaux) for the numerical analysis, R. Veysseyre, D. Weigel and Th. Phan (Ecole Centrale Paris) for the crystallographic part.