An account of the theory of crystallographic groups

Abstract

Introduction. L. Bieberbach in two fundamental papers [2], [3] established the fundamental theorems for the crystallographic groups or Raumgruppen. We propose in this paper to give an almost completely self-contained account of these fundamental facts. We will use only the elementary theory of groups, matrices and polynomials from algebra, the basic geometry of euclidean space and the most elementary topological considerations. At one point we will need the exponential mapping for Lie matrix groups for which various elementary accounts are available. I would like to thank M. Rosenlicht and P. Fong for useful conversations.