On the Abelianization of a Torsion Free Crystallographic Group

Authors

  • Nor'ashiqin Mohd Idrus Department of Mathematics, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, 35900 Tg. Malim, Perak, Malaysia
  • Nor Haniza Sarmin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Hazzirah Izzati Mat Hassim Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Rohaidah Masri Department of Mathematics, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, 35900 Tg. Malim, Perak, Malaysia

DOI:

https://doi.org/10.11113/jt.v70.2424

Keywords:

Torsion free, crystallographic group, Bieberbach group, abelianization

Abstract

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.

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Published

2014-08-27

Issue

Vol. 70 No. 1: September 2014

Section

Science and Engineering

License

Copyright of articles that appear in Jurnal Teknologi belongs exclusively to Penerbit Universiti Teknologi Malaysia (Penerbit UTM Press). This copyright covers the rights to reproduce the article, including reprints, electronic reproductions, or any other reproductions of similar nature.

How to Cite

On the Abelianization of a Torsion Free Crystallographic Group. (2014). Jurnal Teknologi, 70(1). https://doi.org/10.11113/jt.v70.2424