The R∞ property for crystallographic groups of Sol14

We calculate K0 of the rational group algebra of a certain crystallographic group, showing that it contains an element of order 2. We show that this element is the Euler class, and use our calculation to produce a whole family of groups with Euler class of order 2.

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Crystallographic groups with trivial center and outer automorphism group

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000251 ◽

2017 ◽

Vol 164 (2) ◽

pp. 363-368

Author(s):

RAFAŁ LUTOWSKI ◽

ANDRZEJ SZCZEPAŃSKI

Keyword(s):

Automorphism Group ◽

Outer Automorphism ◽

Crystallographic Group ◽

Crystallographic Groups ◽

Outer Automorphism Group ◽

Trivial Center ◽

Cocompact Subgroup ◽

Discrete Cocompact Subgroup

AbstractLet Γ be a crystallographic group of dimension n, i.e. a discrete, cocompact subgroup of Isom(ℝn) = O(n) ⋉ ℝn. For any n ⩾ 2, we construct a crystallographic group with a trivial center and trivial outer automorphism group.

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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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Hexagonal projected symmetries

Acta Crystallographica Section A Foundations and Advances ◽

10.1107/s2053273315012905 ◽

2015 ◽

Vol 71 (5) ◽

pp. 549-558 ◽

Cited By ~ 1

Author(s):

Juliane F. Oliveira ◽

Sofia B. S. D. Castro ◽

Isabel S. Labouriau

Keyword(s):

Pattern Formation ◽

Hexagonal Lattice ◽

Three Dimensional ◽

Dimensional Structure ◽

Crystallographic Group ◽

Two Dimensional ◽

Thin Domains ◽

Crystallographic Groups ◽

Physical Systems ◽

Eye Pattern

In the study of pattern formation in symmetric physical systems, a three-dimensional structure in thin domains is often modelled as a two-dimensional one. This paper is concerned with functions in {\bb R}^{3} that are invariant under the action of a crystallographic group and the symmetries of their projections into a function defined on a plane. A list is obtained of the crystallographic groups for which the projected functions have a hexagonal lattice of periods. The proof is constructive and the result may be used in the study of observed patterns in thin domains, whose symmetries are not expected in two-dimensional models, like the black-eye pattern.

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The algebra of crystallographic groups in dimension two

Journal of Algebra and Its Applications ◽

10.1142/s0219498822500141 ◽

2020 ◽

pp. 2250014

Author(s):

Manuel R. F. Moreira

Keyword(s):

Direct Product ◽

Finite Groups ◽

Natural Generalization ◽

Crystallographic Group ◽

Direct Products ◽

Crystallographic Groups ◽

Extra Effort

Starting with the two semi-direct products of two copies of the group of integers, a description of all possible semi-direct products of these two groups with finite groups, under a faithful action, is obtained. The results are groups that are isomorphic to 16 crystallographic groups in dimension two. An extra effort gives us a group isomorphic to the 17th crystallographic group. Four of the crystallographic groups are shown to be the only solutions to a natural generalization of the semi-direct product of two copies of the group of integers. Finally, we verify that the two additional groups, that are solution to the generalization of the semi-direct product of two copies of the integers, preserve stability in the sense that their semi-direct products with finite groups under a faithful action give rise only to crystallographic groups, if we disregard some trivial outcomes.

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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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