On the abelianization of a torsion free crystallographic group

• Main Authors: Mohd. Idrus, Nor'ashiqin, Sarmin, Nor Haniza, Mat Hassim, Hazzirah Izzati, Masri, Rohaidah
• Format: Article
• Language: English
• Published: Penerbit UTM 2014
• Subjects: TP Chemical technology
• Online Access: http://eprints.utm.my/id/eprint/54313/1/NorHanizaSarmin2014_Ontheabelianizationofatorsion.pdf; http://eprints.utm.my/id/eprint/54313/; http://dx.doi.org/10.11113/jt.v70.2424

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 n-dimensional 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