On the Abelianization of a Torsion Free Crystallographic Group

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

ASM Science Journal ◽

10.32802/asmscj.2020.sm26(1.23) ◽

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

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

10.1063/1.4882560 ◽

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

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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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Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216521500693 ◽

2021 ◽

Author(s):

Oscar Ocampo

Keyword(s):

Fundamental Group ◽

Abelian Subgroup ◽

Holonomy Group ◽

Braid Groups ◽

Crystallographic Group ◽

Flat Manifold ◽

Kahler Geometry ◽

Bieberbach Group ◽

Odd Order ◽

Flat Manifolds

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

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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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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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Almost-Bieberbach groups with (in)finite outer automorphism group

Glasgow Mathematical Journal ◽

10.1017/s0017089500032341 ◽

1998 ◽

Vol 40 (1) ◽

pp. 47-62 ◽

Cited By ~ 3

Author(s):

Wim Malfait ◽

Andrzej Szczepański

Keyword(s):

Automorphism Group ◽

Fundamental Group ◽

Infinite Order ◽

Outer Automorphism ◽

Crystallographic Group ◽

Outer Automorphism Group ◽

Outer Automorphisms ◽

Bieberbach Group ◽

Interesting Class ◽

Almost Bieberbach Group

AbstractIf we investigate symmetry of an infra-nilmanifoldM, the outer automorphism group of its fundamental group (an almost-Bieberbach group) is known to be a crucial object. In this paper, we characterise algebraically almost-Bieberbach groupsEwith finite outer automorphism group Out(E). Inspired by the description of Anosov diffeomorphisms onM, we also present an interesting class of infinite order outer automorphisms. Another possible type of infinite order outer automorphisms arises when comparing Out(E) with the outer automorphism group of the underlying crystallographic group ofE.

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