On some homological functors of a Bieberbach group with symmetric point group

2017 ◽  
Author(s):  
Tan Yee Ting ◽  
Nor’ashiqin Mohd Idrus ◽  
Rohaidah Masri ◽  
Nor Fadzilah Abdul Ladi
Keyword(s):  
Point Group ◽  
Symmetric Point ◽  
Bieberbach Group

THE NONABELIAN TENSOR SQUARE OF A BIEBERBACH GROUP WITH SYMMETRIC POINT GROUP OF ORDER SIX

2015 ◽  
Vol 78 (1) ◽  
Author(s):  
Tan Yee Ting ◽  
Nor'ashiqin Mohd. Idrus ◽  
Rohaidah Masri ◽  
Wan Nor Farhana Wan Mohd Fauzi ◽  
Nor Haniza Sarmin ◽  
...  
Keyword(s):  
Point Group ◽  
Crystallographic Groups ◽  
Symmetric Point ◽  
Bieberbach Group ◽  
Nonabelian Tensor ◽  
Homological Functor ◽  
Polycyclic Groups ◽  
Tensor Square ◽  
Torsion Free

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is given on the Bieberbach groups with symmetric point group of order six. The nonabelian tensor square of a group is a well known homological functor which can reveal the properties of a group. With the method developed for polycyclic groups, the nonabelian tensor square of one of the Bieberbach groups of dimension four with symmetric point group of order six is computed. The nonabelian tensor square of this group is found to be not abelian and its presentation is constructed.

POLYCYCLIC PRESENTATIONS OF THE TORSION FREE SPACE GROUP WITH QUATERNION POINT GROUP OF ORDER EIGHT

2015 ◽  
Vol 77 (33) ◽  
Author(s):  
Siti Afiqah Mohammad ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Free Space ◽  
Point Group ◽  
Space Group ◽  
Bieberbach Group ◽  
Nonabelian Tensor ◽  
Tensor Square ◽  
Torsion Free

A space group of a crystal describes its symmetrical properties. Many mathematical approaches have been explored to study these properties. One of the properties is on exploration of the nonabelian tensor square of the group. Determining the polycyclic presentation of the group before computing the nonabelian tensor square is the method used in this research. Therefore, this research focuses on computing the polycyclic presentations of the torsion free space group named Bieberbach group with a quaternion point group of order eight.

Consistent polycyclic presentation of a Bieberbach group with a nonabelian point group

2016 ◽  
Author(s):  
Siti Afiqah Mohammad ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Bieberbach Group

The homological functor of a Bieberbach group with a cyclic point group of order two

2014 ◽  
Author(s):  
Hazzirah Izzati Mat Hassim ◽  
Nor Haniza Sarmin ◽  
Nor Muhainiah Mohd Ali ◽  
Rohaidah Masri ◽  
Nor'ashiqin Mohd Idrus
Keyword(s):  
Point Group ◽  
Bieberbach Group ◽  
Homological Functor

On the abelianization of all Bieberbach groups of dimension four with symmetric point group of order six

2014 ◽  
Author(s):  
Tan Yee Ting ◽  
Nor'ashiqin Mohd Idrus ◽  
Rohaidah Masri ◽  
Wan Nor Farhana Wan Mohd Fauzi ◽  
Nor Haniza Sarmin ◽  
...  
Keyword(s):  
Point Group ◽  
Symmetric Point

On the Abelianization of a Torsion Free Crystallographic Group

2014 ◽  
Vol 70 (1) ◽  
Author(s):  
Nor'ashiqin Mohd Idrus ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim ◽  
Rohaidah Masri
Keyword(s):  
Fundamental Group ◽  
Riemannian Manifolds ◽  
Point Group ◽  
Crystallographic Group ◽  
Finite Point ◽  
Lattice Group ◽  
Bieberbach Group ◽  
Trivial Center ◽  
Free Abelian Groups ◽  
Torsion Free

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.

The presentation of the nonabelian tensor square of a Bieberbach group of dimension five with dihedral point group

2014 ◽  
Author(s):  
Wan Nor Farhana Wan Mohd Fauzi ◽  
Nor'ashiqin Mohd Idrus ◽  
Rohaidah Masri ◽  
Tan Yee Ting ◽  
Nor Haniza Sarmin ◽  
...  
Keyword(s):  
Point Group ◽  
Bieberbach Group ◽  
Nonabelian Tensor ◽  
Tensor Square

scholarly journals The Exterior Square of a Bieberbach Group with Quaternion Point Group of Order Eight

2020 ◽  
pp. 1-7
Author(s):  
S.A. Mohammad ◽  
N.H. Sarmin ◽  
H.I. Mat Hassim
Keyword(s):  
Mathematical Method ◽  
Point Group ◽  
Mathematical Structure ◽  
Mathematical Representation ◽  
Crystallographic Group ◽  
Finite Point ◽  
Bieberbach Group ◽  
Abelian Lattice ◽  
Torsion Free ◽  
Exterior Square

A Bieberbach group is defined to be a torsion free crystallographic group which is an extension of a free abelian lattice group by a finite point group. This paper aims to determine a mathematical representation of a Bieberbach group with quaternion point group of order eight. Such mathematical representation is the exterior square. Mathematical method from representation theory is used to find the exterior square of this group. The exterior square of this group is found to be nonabelian. Keywords: mathematical structure; exterior square; Bieberbach group; quaternion point group

The central subgroup of the nonabelian tensor square of Bieberbach group of dimension three with point group C2 × C2

2017 ◽  
Author(s):  
Nor Fadzilah Abdul Ladi ◽  
Rohaidah Masri ◽  
Nor’ashiqin Mohd Idrus ◽  
Tan Yee Ting
Keyword(s):  
Point Group ◽  
Bieberbach Group ◽  
Nonabelian Tensor ◽  
Tensor Square
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