Clasificación de grupos cristalográficos planos

Abstract

Crystallographic groups are groups whose elements are isometries of a metric space. These groups are 17 in the Euclidean plane and we can find them in nature or tilings. Many scientists study these groups in fields other than Mathematics, such as Chemistry. Throughout history, some people have given the list of 17 groups, and have been interested in the study of them in higher dimensions. In this work there are two different blocks. In the first one, we recall some concepts of group theory. Then we explain how a group acts on the elements of a set. Finally we recall some notions of Euclidean plane to explain the isometries of the plane. The second part of this work focuses on an article of Schwarzenberger [14]. We show what is a crystallographic group and we prove that there are exactly 17 of them. To do that, we divide the groups into three types: those which contain, respectively, no reflections, exactly one or more than one. In addition, we have included pictures for an easier understanding of the text, and examples of each group.