On the Abelianization of a Torsion Free Crystallographic Group

A torsion free crystallographic group, which is also known as a bieberbach group is a generalization of free abelian groups. it is an extension of a lattice group by a finite point group. the study of ndimensional crystallographic group had been done by many researchers over a hundred years ago. a bieberbach group has been characterized as a fundamental group of compact, connected, flat riemannian manifolds. in this paper, we characterize bieberbach groups with trivial center as exactly those with finite abelianizations. the abelianization of a bieberbach group is shown to be finite if the center of the group is trivial.