Burckhardtian determination of the space groups. II.
type: Item
instance of: scholarly article
title: Burckhardtian determination of the space groups. II.
zbMATH Open document ID: 1062.20054
author: Ralph Strebel
published in: Elemente der Mathematik
publication date: 2004-09-22T00:00:00Z
review text: The central concept of part II is given by the congruence introduced by Frobenius in 1911 for characterizing the properties of systems of nonprimitive translations in space groups. The Frobenius congruences also play an important role in the Burckhardtian derivation of the 3-dimensional space groups. These are in fact crossed homomorphisms with an equivalence relation given by principal crossed homomorphisms. This leads to the first cohomology group of integral matrix groups (representatives of the arithmetic classes discussed in part I [see the preceding review Zbl 1062.20053]) with values in \(\mathbb{R}/\mathbb{Z}\) and to the algorithm of Zassenhaus (1948) for the derivation of space groups. For getting the affine inequivalent space groups, which also are non-isomorphic as abstract groups, one needs the normalizer of these first cohomology groups. The 3-dimensional tetragonal space groups are used as illustrative example.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 20-03
Mathematics Subject Classification ID: 01A60
Mathematics Subject Classification ID: 52C23
zbMATH DE Number: 2102273
zbMATH Keywords: space groups
zbMATH Keywords: Frobenius congruences
zbMATH Keywords: Zassenhaus algorithm
reviewed by: A. Janner
describes a project that uses: CARAT
MaRDI profile type: Publication
full work available at URL: s00017-001-0207-y
OpenAlex ID: W2096030983
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DOI: 10.1007/S00017-001-0207-Y
description: scientific article; zbMATH DE number 2102273