Author/Authors :
Idrus, Nor’ashiqin Mohd Universiti Pendidikan Sultan Idris - Faculty of Science and Mathematics - Department of Mathematics, Malaysia , Sarmin, Nor Haniza Universiti Teknologi Malaysia - Faculty of Science - Department of Mathematical Sciences, Malaysia , Mat Hassim, Hazzirah Izzati Universiti Teknologi Malaysia - Faculty of Science - Department of Mathematical Sciences, Malaysia , Masri, Rohaidah Universiti Pendidikan Sultan Idris - Faculty of Science and Mathematics - Department of Mathematics, Malaysia
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 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.
Keywords :
Torsion free , crystallographic group , Bieberbach group , abelianization
