Formulation of the Nonabelian Tensor Square of a Bieberbach Group

doi:10.35940/ijrte.b1046.0782s219

Formulation of the Nonabelian Tensor Square of a Bieberbach Group

International Journal of Recent Technology and Engineering - 2 ◽

10.35940/ijrte.b1046.0782s219 ◽

2019 ◽

Vol 8 (2S2) ◽

pp. 260-263

Keyword(s):

Bieberbach Group ◽

Nonabelian Tensor ◽

Point Set ◽

Tensor Square ◽

Torsion Free

A Bieberbach set can be categorized as a torsion free crystallographic set. Some properties can be explored from the set such as the property of nonabelian tensor square. The nonabelian tensor square is one type of the homological factors of sets. This paper focused on a Bieberbach set with C2 ×C2 as the point set of lowest dimension three. The purpose of this paper is to determine the generalization of the formula of the nonabelian tensor square of one Bieberbach set with point set C2 × C2of lowest dimension three which is denoted by S2 (3) up to dimensionn. The polycyclic presentation, the abelianization of S2 (3) and the central subgroup of the nonabelian tensor square of S2 (3) are also presented.

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

Jurnal Teknologi ◽

10.11113/jt.v77.7020 ◽

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.

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The Exterior Squares of Some Crystallographic Groups

Jurnal Teknologi ◽

10.11113/jt.v62.1882 ◽

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.

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

Jurnal Teknologi ◽

10.11113/jt.v78.4385 ◽

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.

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On the Generalization of the Nonabelian Tensor Square of a Bieberbach Group with Symmetric Point Group

Indian Journal of Science and Technology ◽

10.17485/ijst/2016/v9is1/92736 ◽

2016 ◽

Vol 9 (S1) ◽

Cited By ~ 1

Author(s):

Y. T. Tan ◽

N. MohdIdrus ◽

R. Masri ◽

N. H. Sarmin ◽

H. I. Mat Hassim

Keyword(s):

Point Group ◽

Symmetric Point ◽

Bieberbach Group ◽

Nonabelian Tensor ◽

Tensor Square

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

Jurnal Teknologi ◽

10.11113/jt.v79.10677 ◽

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.

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