 The fourth step and Poincaré’s main result is that each set of displacement classes (external &amp; internal) forms a group in the mathematical sense. The general concept of a group is a form of our understanding that “pre-exists in our mind” (1902:107; 2017: 56 and 68; 1898: 42–43) or is an ante rem structure, which the specific displacement-structure \((=\) transformation group) exemplifies. Thus, the genesis of geometry is based on an epistemological process founded on previous classifications, carried out as a relationship between a structure as norm of invariance and conventionally adapted systems as instantiations or exemplifications of these norms.