Decomposition of Sohncke space groups into products of Bieberbach and symmorphic parts

2015 ◽  
Vol 230 (12) ◽  
Author(s):  
Gregory S. Chirikjian ◽  
Kushan Ratnayake ◽  
Sajdeh Sajjadi
Keyword(s):  
Direct Product ◽  
Free Space ◽  
Special Class ◽  
The Other ◽  
Pure Point ◽  
Direct Products ◽  
Space Groups ◽  
Point Transformations ◽  
Point Groups ◽  
Torsion Free

AbstractPoint groups consist of rotations, reflections, and roto-reflections and are foundational in crystallography. Symmorphic space groups are those that can be decomposed as a semi-direct product of pure translations and pure point subgroups. In contrast, Bieberbach groups consist of pure translations, screws, and glides. These “torsion-free” space groups are rarely mentioned as being a special class outside of the mathematics literature. Every space group can be thought of as lying along a spectrum with the symmorphic case at one extreme and Bieberbach space groups at the other. The remaining nonsymmorphic space groups lie somewhere in between. Many of these can be decomposed into semi-direct products of Bieberbach subgroups and point transformations. In particular, we show that those 3D Sohncke space groups most populated by macromolecular crystals obey such decompositions. We tabulate these decompositions for those Sohncke groups that admit such decompositions. This has implications to the study of packing arrangements in macromolecular crystals. We also observe that every Sohncke group can be written as a product of Bieberbach and symmorphic subgroups, and this has implications for new nomenclature for space groups.

The crystallographic point groups as semi-direct products

1963 ◽  
Vol 255 (1054) ◽  
pp. 216-240 ◽  
Keyword(s):  
Direct Product ◽  
Invariant Subgroup ◽  
Projection Operators ◽  
Direct Products ◽  
Finite Rotations ◽  
Space Groups ◽  
Symmetry Adapted Functions ◽  
Point Groups ◽  
Definition Of

This paper aims at providing a systematic treatment of the crystallographic point groups. Some well-known properties of them, in terms of the theory of the poles of finite rotations, are first discussed, so as to provide a simple way for recognizing their invariant subgroups. A definition of the semi-direct product is then given, and it is shown that all crystallographic point groups can be expressed as a semi-direct product of one of their invariant subgroups by a cyclic subgroup. Many useful relations between point groups can be obtained by exploiting the properties of the triple and mixed triple semi-direct products, which are defined. Much of the rest of the paper is devoted to the theory of the representations of semi-direct products. The treatment here parallels that given by Seitz (1936) for the reduction of space groups in terms of the representations of its invariant subgroups (the translation groups). The latter, however, are always Abelian and this is not always the case for point groups. The full treatment of the general case, such as given by McIntosh (1958), is laborious and it is shown that, if the emphasis is placed on the bases of the representations, rather than the representations themselves, it is possible to achieve the reduction of the point groups by a method hardly more involved than that required when the invariant subgroup is Abelian. It is also shown that, just as for space groups, the representations of the invariant subgroups can be denoted and visualized by means of a vector, which allows a very rapid classification of the representations, very much as the k vector as used by Bouckaert, Smoluchowski & Wigner (1936) allows the formalism of the Seitz method for space groups to be carried out in a graphical fashion. One of the major consequences of this work is that it affords a substantial simplification in the use of the symmetrizing and projection operators that are required to obtain symmetry-adapted functions: a very systematic alternative to the method given by Melvin (1956) is therefore provided. In the last section of the paper all the techniques discussed are applied in detail, as an example, to the cubic groups. The projection operators are used to obtain symmetry-adapted spherical harmonics for these groups. The paper might be found useful as an introduction to the methods for the reduction of space groups.

Classes d'éléments conjugués des groupes cristallographiques

1978 ◽  
Vol 34 (6) ◽  
pp. 895-900
Author(s):  
J. Sivardière
Keyword(s):  
Finite Group ◽  
Double Point ◽  
Invariant Subgroup ◽  
Factor Group ◽  
Direct Products ◽  
Simple Point ◽  
Space Groups ◽  
Point Groups ◽  
A Finite Group

Let G be a finite group, H an invariant subgroup and F the corresponding factor group. The classes of conjugated elements of G are derived from the classes of H and F. We consider simple point groups and symmorphic space groups, which are semi-direct products H^F, then double point groups and non- symmorphic space groups, which are extensions of F by H.

scholarly journals Hyperbolicity of Direct Products of Graphs

Symmetry ◽  
2018 ◽  
Vol 10 (7) ◽  
pp. 279 ◽  
Author(s):  
Walter Carballosa ◽  
Amauris de la Cruz ◽  
Alvaro Martínez-Pérez ◽  
José Rodríguez
Keyword(s):  
Direct Product ◽  
Bipartite Graphs ◽  
The Other ◽  
Necessary Condition ◽  
Direct Products ◽  
Product Graphs ◽  
Hyperbolicity Constant

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).

TEST ELEMENTS IN DIRECT PRODUCTS OF GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 751-756 ◽  
Author(s):  
JOHN C. O'NEILL ◽  
EDWARD C. TURNER
Keyword(s):  
Direct Product ◽  
Commutator Subgroup ◽  
Free Groups ◽  
Surface Groups ◽  
Direct Products ◽  
Products Of Groups ◽  
Test Elements ◽  
Torsion Free

We characterize test elements in the commutator subgroup of a direct product of certain groups in terms of test elements of the factors. This provides explicit examples of test elements in direct products whose factors are free groups or surface groups and a tool for doing the same for torsion free hyperbolic factors.

scholarly journals Torsion-Free Space Groups, 2-Cohomology, and Scott Modules

1996 ◽  
Vol 180 (3) ◽  
pp. 889-896
Author(s):  
Gerald Cliff ◽  
Hongliu Zheng
Keyword(s):  
Free Space ◽  
Space Groups ◽  
Torsion Free

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

scholarly journals ON NEAR-RING IDEMPOTENTS AND POLYNOMIALS ON DIRECT PRODUCTS OF Ω-GROUPS

2001 ◽  
Vol 44 (2) ◽  
pp. 379-388 ◽  
Author(s):  
Erhard Aichinger
Keyword(s):  
Direct Product ◽  
Subject Classification ◽  
Idempotent Element ◽  
Direct Products ◽  
Primary 16Y30 ◽  
Secondary 08A40

AbstractLet $N$ be a zero-symmetric near-ring with identity, and let $\sGa$ be a faithful tame $N$-group. We characterize those ideals of $\sGa$ that are the range of some idempotent element of $N$. Using these idempotents, we show that the polynomials on the direct product of the finite $\sOm$-groups $V_1,V_2,\dots,V_n$ can be studied componentwise if and only if $\prod_{i=1}^nV_i$ has no skew congruences.AMS 2000 Mathematics subject classification: Primary 16Y30. Secondary 08A40

scholarly journals ALGEBRA* VIOLENCE ALGEBRA CRATY

2020 ◽  
pp. 12-23
Author(s):  
Vadym Chuiko ◽  
Valerii Atamanchuk-Angel
Keyword(s):  
Sustainable Development ◽  
Direct Product ◽  
The State ◽  
State System ◽  
The Other ◽  
Philosophical Thought ◽  
Human Culture ◽  
Full Responsibility ◽  
Abstract Notion ◽  
Almost All

Almost all philosophy about the state system has concentrated on the authorities. Any function of the state can be represented as a superposition of the functions of violence / coercion. Ultimately, the state appears to be a kind of plurality of subjects with a definite crater power / coercion / violence operation on it. The algebra of trust on the multiplicity of owners of themselves, endowed with free future, is each of them is only a part of nature, еру carrier of the part of the general human culture, and for their completeness, they have and understand the need for the Other. This is the philosophy of solving political, environmental, and climate challenges not through violent / voluntaristic methods, but by the recognition of sovereign rights and the search for ways to achieve sustainable development. Any cracy / power / coercion / violence must be separated from the models of society, the state. Public agreement is not an agreement with the abstract notion of the state, but an agreement with definite elected people who have gained the trust of those to whom they temporarily render their services. Contract is temporary, limited by period, with obligatory full responsibility of the parties. Scientific novelty. For more than two thousand years, long before Aristotle and Plato, European philosophical thought, reflecting on the structure of society, wanders in the labyrinths of kratia. Modern achievements of mathematics provide an opportunity to build ideal political objects, and a direct product of material and ideal government building. (Example of a trust algebra [4].)

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