The Exterior Squares of Some Crystallographic Groups

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

Journal of Physics Conference Series ◽

10.1088/1742-6596/893/1/012006 ◽

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

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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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On the Abelianization of a Torsion Free Crystallographic Group

Jurnal Teknologi ◽

10.11113/jt.v70.2424 ◽

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.

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Birational invariants of crystals and fields with a finite group of operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100068717 ◽

1990 ◽

Vol 107 (3) ◽

pp. 417-424 ◽

Cited By ~ 2

Author(s):

Daniel R. Farkas

Keyword(s):

Finite Group ◽

Integral Representation ◽

Point Group ◽

Crystallographic Group ◽

Crystallographic Groups ◽

Commutative Domain ◽

A Finite Group ◽

Torsion Free ◽

Maximal Subfield

It is well known that an n-dimensional crystallographic group can be reconstructed from its point group, the integral representation of the point group which arises from its action on the translation lattice, and the 2-cocycle which glues the point group to the lattice ([2]). In practice, this constitutes a complicated list of invariants. When confronted with the classification of objects possessing a rich structure, the algebraic geometer first attempts to find more coarse birational invariants. We begin such a programme for torsion-free crystallographic groups. More precisely, if Γ is a torsion-free crystallographic group and k is a field then the group algebra k[Γ] is a non-commutative domain (see [6], chapter 13). It can be localized at its centre to yield a division algebra k(Γ) which is a crossed product; the Galois group is the point group and it acts on the rational function field generated by k and the lattice (regarded multiplicatively), which is a maximal subfield ([3]). What are thecommon invariants of Γ1 and Γ2 when k(Γ1) and k(Γ2) are isomorphic k-algebras?

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