1. Introduction

This paper deals with the classification of discrete subgroups of the Euclidean group (the group of isometries of the Euclidean plane, denoted as E₂). We insist on the structure of the Euclidean group and its subgroups.

We tried, whenever it was possible, to avoid the transition to the synthetic geometry (this principle was not respected when the geometrical arguments were much more simple and elegant than analytical or group theory arguments).

The paper needs the basic knowledge of group theory (axioms, subgroups, normal subgroups, isomorphism), linear algebra (vectors, matrix operations), analytic geometry (coordinates), and general function properties. Some less known notions of group theory are introduced in Appendix.

Unless the opposite is emphasized, we used the standard notation.