scholarly journals Polycyclic transformations of crystallographic groups with quaternion point group of order eight

2017 ◽  
Vol 13 (4) ◽  
pp. 788-791
Author(s):  
Siti Afiqah Mohammad ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Crystallographic Groups ◽  
Nonabelian Tensor ◽  
Homological Invariants ◽  
Tensor Square

Exploration of a group's properties is vital for better understanding about the group.  Amongst other properties, the homological invariants including the nonabelian tensor square of a group can be explicated by showing that the group is polycyclic.  In this paper, the polycyclic presentations of certain crystallographic groups with quaternion point group of order eight are shown to be consistent; which implies that these groups are polycyclic.

scholarly journals THE CENTRAL SUBGROUPS OF THE NONABELIAN TENSOR SQUARES OF SOME BIEBERBACH GROUPS WITH ELEMENTARY ABELIAN 2-GROUP POINT GROUP

2017 ◽  
Vol 79 (7) ◽  
Author(s):  
Nor Fadzilah Abdul Ladi ◽  
Rohaidah Masri ◽  
Nor'ashiqin Mohd Idrus ◽  
Nor Haniza Sarmin ◽  
Tan Yee Ting
Keyword(s):  
Point Group ◽  
Crystallographic Groups ◽  
Nonabelian Tensor ◽  
Tensor Square ◽  
Torsion Free

Bieberbach groups are torsion free crystallographic groups. In this paper, our focus is on the Bieberbach groups with elementary abelian 2-group point group,  The central subgroup of the nonabelian tensor square of a group  is generated by  for all  in  The purpose of this paper is to compute the central subgroups of the nonabelian tensor squares of two Bieberbach groups with elementary abelian 2-point group of dimension three. 

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.

scholarly journals The nonabelian tensor square of a crystallographic group with quaternion point group of order eight

2017 ◽  
Vol 893 ◽  
pp. 012006
Author(s):  
Siti Afiqah Mohammad ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Crystallographic Group ◽  
Nonabelian Tensor ◽  
Tensor Square

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.

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

The Exterior Squares of Some Crystallographic Groups

2013 ◽  
Vol 62 (3) ◽  
Author(s):  
Hazzirah Izzati Mat Hassim ◽  
Nor Haniza Sarmin ◽  
Nor Muhainiah Mohd Ali ◽  
Rohaidah Masri Masri ◽  
Nor’ashiqin Mohd Idrus Mohd Idrus
Keyword(s):  
Quotient Space ◽  
Discrete Subgroup ◽  
Factor Group ◽  
Crystallographic Group ◽  
Crystallographic Groups ◽  
Bieberbach Group ◽  
Nonabelian Tensor ◽  
Point Groups ◽  
Tensor Square ◽  
Torsion Free

A crystallographic group is a discrete subgroup G of the set of isometries of Euclidean space En, where the quotient space En/G is compact. A specific type of crystallographic groups is called Bieberbach groups. A Bieberbach group is defined to be a torsion free crystallographic group. In this paper, the exterior squares of some Bieberbach groups with abelian point groups are computed. The exterior square of a group is the factor group of the nonabelian tensor square with the central subgroup of the 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

scholarly journals SOME CONSIDERATIONS ON THE NONABELIAN TENSOR SQUARE OF CRYSTALLOGRAPHIC GROUPS

2011 ◽  
Vol 04 (02) ◽  
pp. 271-282 ◽  
Author(s):  
Ahmad Erfanian ◽  
Francesco G. Russo ◽  
Nor Haniza Sarmin
Keyword(s):  
Tensor Product ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Schur Multiplier ◽  
Crystallographic Groups ◽  
Nonabelian Tensor ◽  
Tensor Square

The nonabelian tensor square G ⊗ G of a polycyclic group G is a polycyclic group and its structure arouses interest in many contexts. The same assertion is still true for wider classes of solvable groups. This motivated us to work on two levels in the present paper: on a hand, we investigate the growth of the Hirsch length of G ⊗ G by looking at that of G, on another hand, we study the nonabelian tensor product of pro–p–groups of finite coclass, which are a remarkable class of solvable groups without center, and then we do considerations on their Hirsch length. Among other results, restrictions on the Schur multiplier will be discussed.

The nonabelian tensor square of Bieberbach group of dimension five with dihedral point group of order eight

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