Homological functor of a torsion free crystallographic group of dimension five with a nonabelian point group

2014 ◽  
Author(s):  
Tan Yee Ting ◽  
Nor'ashiqin Mohd. Idrus ◽  
Rohaidah Masri ◽  
Nor Haniza Sarmin ◽  
Hazzirah Izzati Mat Hassim
Keyword(s):  
Point Group ◽  
Crystallographic Group ◽  
Homological Functor ◽  
Torsion Free

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.

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

1990 ◽  
Vol 107 (3) ◽  
pp. 417-424 ◽  
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?

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 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.

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 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

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.

scholarly journals Orbits of crystallographic embedding of non-crystallographic groups and applications to virology

2015 ◽  
Vol 71 (6) ◽  
pp. 569-582 ◽  
Author(s):  
Reidun Twarock ◽  
Motiejus Valiunas ◽  
Emilio Zappa
Keyword(s):  
Dynamic Properties ◽  
Group Structure ◽  
Point Group ◽  
Theoretical Method ◽  
Lattice Points ◽  
Crystallographic Group ◽  
Icosahedral Symmetry ◽  
Point Sets ◽  
Finite Structures ◽  
Crystallographic Symmetry

The architecture of infinite structures with non-crystallographic symmetries can be modelledviaaperiodic tilings, but a systematic construction method for finite structures with non-crystallographic symmetry at different radial levels is still lacking. This paper presents a group theoretical method for the construction of finite nested point sets with non-crystallographic symmetry. Akin to the construction of quasicrystals, a non-crystallographic groupGis embedded into the point group {\cal P} of a higher-dimensional lattice and the chains of allG-containing subgroups are constructed. The orbits of lattice points under such subgroups are determined, and it is shown that their projection into a lower-dimensionalG-invariant subspace consists of nested point sets withG-symmetry at each radial level. The number of different radial levels is bounded by the index ofGin the subgroup of {\cal P}. In the case of icosahedral symmetry, all subgroup chains are determined explicitly and it is illustrated that these point sets in projection provide blueprints that approximate the organization of simple viral capsids, encoding information on the structural organization of capsid proteins and the genomic material collectively, based on two case studies. Contrary to the affine extensions previously introduced, these orbits endow virus architecture with an underlying finite group structure, which lends itself better to the modelling of dynamic properties than its infinite-dimensional counterpart.

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