Several new results on quasicrystallographic groups in Novikov's sense
type: Item
instance of: scholarly article
title: Several new results on quasicrystallographic groups in Novikov's sense
zbMATH Open document ID: 0723.20020
DOI: 10.1007/BF01157439
author: S. A. Piunikhin
published in: Mathematical Notes
publication date: 1990-01-01T00:00:00Z
review text: A finitely generated abelian subgroup \(T\subset R^ k\), which generates \(R^ k\) as a linear space is called a quasilattice in R. A subgroup G of the group \(E_ k\) of all isometries of k-dimensional Euclidean space is called k-dimensional quasi-crystallographic group, iff its intersection with the subgroup \(R^ k\subset E_ k\) of all translations is some quasilattice \(T\subset R^ k.\) These groups can contain infinite-order rotations even in dimension two. It is proved that in dimension three these groups can be even infinitely generated. The problem of classification (up to group-theoretical isomorphism) of quasi-crystallographic groups with finite group of linear parts is reduced to problems of representation theory of finite groups. In the majority of particular cases in dimension three these algebraic problems are solved.
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 4191992
zbMATH Keywords: quasilattice
zbMATH Keywords: translations
zbMATH Keywords: infinitely generated
zbMATH Keywords: quasi-crystallographic groups
reviewed by: Sergey Piunikhin
MaRDI profile type: Publication
cites work: Q4288966
cites work: Integral Representations of Dihedral Groups of Order 2p
cites work: Q4046233
cites work: Q3681011
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description: scientific article; zbMATH DE number 4191992