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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{crystallographic group} [[!redirects space group]] \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory} [[!include group theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{Classification}{Classification}\dotfill \pageref*{Classification} \linebreak \noindent\hyperlink{CompactFlatOrbifolds}{Compact flat orbifolds}\dotfill \pageref*{CompactFlatOrbifolds} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \textbf{space group} in [[dimension]] $n$, also known as a \textbf{crystallographic group}, is a [[subgroup]] of the corresponding [[Euclidean group]], hence of the [[isometry group]] of [[Euclidean space]] $\mathbb{R}^n$, that contains some [[lattice in a vector space|lattice]] in $\mathbb{R}^n$ as a [[subgroup]], and is contained within the [[automorphism group]] of that lattice. In other words, it is a subgroup of the automorphism group of the lattice that contains all the [[translations]] by elements of the lattice itself. Equivalently, a crystallographic group on a [[Euclidean space]] $E$ is a [[finite group|finite]] [[subgroup]] $S \subset Iso(E)$ of the [[isometry group]] of $E$ (its [[Euclidean group]]) that contains a [[lattice (discrete subgroup)|lattice]] $N \subset E \subset Iso(E)$ of [[translation group|translations]] as a [[normal subgroup]] $N \subset S$. The corresponding [[quotient group]] $G \coloneqq S/N$ is called the \emph{[[point group]]} of the crystallographic group. This situation is reflected in [[short exact sequences]] \begin{displaymath} \itexarray{ & 1 && 1 \\ & \downarrow && \downarrow \\ {\text{normal subgroup} \atop \text{lattice of translations}} & N &\subset& E & {\text{translation} \atop \text{group}} \\ & \big\downarrow && \big\downarrow \\ {\text{crystallographic} \atop \text{group}} & S &\subset& Iso(E) & {\text{Euclidean} \atop \text{isometry group}} \\ & \big\downarrow && \big\downarrow \\ {\text{point} \atop \text{group}} & G &\subset& O(E) & {\text{orthogonal} \atop \text{group}} \\ & \downarrow && \downarrow \\ & 1 && 1 } \end{displaymath} If the [[short exact sequence]] on the left [[split exact sequence|splits]], hence if the space group $S \simeq G \ltimes N$ is the [[semidirect product]] of the [[point group]] with the translational lattice, $S$ is called a \emph{symmorphic space group}. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{Classification}{}\subsubsection*{{Classification}}\label{Classification} In 2 [[dimensions]], there are precisely 17 crystallographic groups, which are distinct up to [[isomorphism]]; these are known as the \emph{[[wallpaper groups]]}. In 3 dimensions, there are 219 distinct types, or 230 if chiral copies are considered distinct. The classification of space groups has been carried out up to 6 dimensions. On the classification of symmorphic space groups see also \href{https://mathoverflow.net/q/77682/381}{this MO comment}. From \hyperlink{ChuprunovKuntsevich88}{Chuprunov-Kuntsevich 88}: \begin{quote}% Let us make a brief survey of the main achievements in the $n$-dimensional crystallography that have been amply covered in the literature on the subject 15-17. When Fedorov and Sehoenflies had completed the derivation of 230 space group types of crystals it was natural to consider a possibility of derivation of corresponding groups in higher dimensions. In 1911-12 Bieberbach and Frobenius developed a general theory of the group symmetry of the n-dimensional lattices and proved the existence of a finite number of nonisomorphous space groups in the n-dimensional Euclidean space with an arbitrary number of n. Basing on this general theory, in 1948 Zassenhaus suggested an algorithm to derive the n-dimensional space groups as extensions of the translation subgroups of these groups using point groups. About 1950 Hermann gave a complete description of the possible crystallographic symmetry operations in higher dimensions and discussed the lattices of maximal symmetry and their crystal classes. In 1951 Hurley found 222 geometric crystal classes in the four-dimensional Euclidean space making use of the 1889 work by Goursat who had enumerated the classes of finite groups of the real 4 × 4 matrixes. Later this number was corrected to 227. At present classification of crystallographic groups in the four-dimensional Euclidean space is completed in the main. A complete list of 4783 types of four-dimensional space groups was computed in 1973 and given in an excellent monograph \hyperlink{BrownBulowNeubuserWondratschekZassenhaus78}{``Crystallographic groups of four- dimensional space'' by Brown et al.} 15. These groups were derived on the base of the nine maximal arithmetic crystal classes, derived by Dade in 1965, which allowed one to determine all of the 710 four-dimensional arithmetic classes and to calculate the normalizers of finite groups of the unimodular 4 x 4 matrixes needed for the Zassenhaus algorithm. The monograph 15 is of interest not only by having a complete description of all classes of the four-dimensional crystallographic groups but also by taking a deeper approach to the system of classification of the n-dimensional crystallographic groups, as well as by giving characteristic properties of the four-dimensional crystallographic groups in comparison with that in lower dimensions. One of these properties is enantiomorphizm exhibited not only by the space group types but also by Bravais types of lattices, arithmetic classes and geometric classes. For the first time this phenomenon was found by Shtogrin 18. The n-dimensional mathematical crystallography is still in progess. Ryshkov 19 determined all maximal arithmetic crystal classes of five-dimensional Euclidean space. Some categories of five- and six-dimensional ``small'' groups isomorphic to the three-dimensional groups of symmetry, anti- symmetry, two-fold antisymmetry, p- and p'-symmetry were derived by Palistrant 20. Some aspects of the mathematical theory applied to the n-dimensional crystallography were considered 15, 21. \end{quote} \hypertarget{CompactFlatOrbifolds}{}\subsubsection*{{Compact flat orbifolds}}\label{CompactFlatOrbifolds} \begin{prop} \label{InducedPointGroupActionOnTorus}\hypertarget{InducedPointGroupActionOnTorus}{} \textbf{(induced point group action on torus)} The assumption that the crystallographic translation group $N \subset S$ is a [[normal subgroup]] \begin{displaymath} 1 \to N \longrightarrow S \longrightarrow G \to 1 \end{displaymath} implies that the [[action]] of the [[point group]] $G = S/N$ descends to the [[torus]] [[quotient space]] $E/N$ \begin{displaymath} \itexarray{ E &\overset{g}{\longrightarrow}& E \\ \big\downarrow && \big\downarrow \\ E/N &\underset{g}{\longrightarrow}& E/N } \end{displaymath} \end{prop} \begin{proof} By the definition of [[quotient space]], the condition for this to be the case is that for all $x \in E$ we have $g(n(x)) = n'(g(x))$, or equivalently $g(n(g^{-1}(y))) = n'(y)$, which is implied by $N$ being a [[normal subgroup]]: $g N g^{-1} = N$. \end{proof} \begin{remark} \label{}\hypertarget{}{} The further [[homotopy quotient]] $(E/N)\sslash G$ of the [[torus]] $E/N$ by this induced [[action]] of the [[point group]] $G$ is a [[compact topological space|compact]] [[flat orbifold]], and most compact flat orbifolds arise this way. \end{remark} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item \emph{The Crystallographic Groups}, Pure and Applied Mathematics Volume 50, 1972, Pages 16-60 () \item H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, \emph{Crystallographic Groups of Four-Dimensional Space}, John Wiley, New York, 1978. \item Daniel R. Farkas, \emph{Crystallographic groups and their mathematics}, Rocky Mountain J. Math. Volume 11, Number 4 (1981), 511-552 (\href{https://projecteuclid.org/euclid.rmjm/1250128489}{doi:10.1216/RMJ-1981-11-4-511}) \item E. V. Chuprunov, T. S. Kuntsevich, \emph{$n$-Dimensional space groups and regular point systems}, Comput. Math. Applic. Vol. 16, No. 5-8, pp. 537-543, 1988 () \item D. Weigel, T. Phan and R. Veysseyre, \emph{Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in four-dimensional space}, Acta Cryst. (1987). A43, 294-304 (\href{https://doi.org/10.1107/S0108767387099367}{doi:10.1107/S0108767387099367}) \item [[GAP]] package, \emph{The Crystallographic Groups Catalog} (\href{http://www.math.rwth-aachen.de/~Greg.Gamble/gap4r3/pkg/crystcat/htm/CHAP001.htm}{web}) \end{itemize} See also \begin{itemize}% \item [[eom]], \emph{\href{https://www.encyclopediaofmath.org/index.php/Crystallographic_group}{Crystallographic group}} \item Groupprops, \emph{\href{https://groupprops.subwiki.org/wiki/Space_group}{Space group}} \item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Space_group}{Space group}} \item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Point_group}{Point group}} \end{itemize} [[!redirects space groups]] [[!redirects point group]] [[!redirects point groups]] [[!redirects symmorphic space group]] [[!redirects symmorphic space groups]] [[!redirects symmorphic crystallographic group]] [[!redirects symmorphic crystallographic groups]] [[!redirects crystallographic group]] [[!redirects crystallographic groups]] [[!redirects representation torus]] [[!redirects representation tori]] \end{document}