Contents

group theory

# Contents

## Idea

A space group in dimension $n$, also known as a 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 $\mathbb{R}^n$ 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 subgroup $S \subset Iso(E)$ of the isometry group of $E$ (its Euclidean group) that contains a lattice $N \subset E \subset Iso(E)$ of translations as a normal subgroup $N \subset S$, such that the corresponding quotient group, called the point group of the crystallographic group, is a subgroup $G \coloneqq S/N \;\subset\; O(E)$ of the orthogonal group.

In short, a crystallographic groups is exhibited by an inclusion of short exact sequences of (non-abelian) groups, as follows:

$\array{ & 1 && 1 \\ & \downarrow && \downarrow \\ {\text{normal subgroup} \atop \text{lattice of translations}} & N &\subset& E & {\text{translation} \atop \text{group}} \\ & \big\downarrow && \big\downarrow \\ {\text{crystallographic} \atop \text{group}} & S &\subset& Iso(E) & {\text{Euclidean} \atop \text{isometry group}} \\ & \big\downarrow && \big\downarrow \\ {\text{point} \atop \text{group}} & G &\subset& O(E) & {\text{orthogonal} \atop \text{group}} \\ & \downarrow && \downarrow \\ & 1 && 1 }$

(This perspective on crystallographic groups is known as Bieberbach's first theorem, see for instance Tolcachier 19, Theorem 2.3, also Freed-Moore 13, (0.2).)

If the short exact sequence on the left 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 symmorphic space group.

## Properties

### Classification

In 2 dimensions, there are precisely 17 crystallographic groups, which are distinct up to isomorphism; these are known as the 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 this MO comment.

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 “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].

### Compact flat orbifolds

###### Proposition

(induced point group action on torus)

The assumption that the crystallographic translation group $N \subset S$ is a normal subgroup

$1 \to N \longrightarrow S \longrightarrow G \to 1$

implies that the action of the point group $G = S/N$ descends to the torus quotient space $E/N$

$\array{ E &\overset{g}{\longrightarrow}& E \\ \big\downarrow && \big\downarrow \\ E/N &\underset{g}{\longrightarrow}& E/N }$
###### 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: $g N g^{-1} = N$.

###### Remark

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 flat orbifold, and most compact flat orbifolds arise this way.

## References

• The Crystallographic Groups, Pure and Applied Mathematics Volume 50, 1972, Pages 16-60 (doi:10.1016/S0079-8169(08)60959-9)

• H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space, John Wiley, New York, 1978.

• Daniel R. Farkas, Crystallographic groups and their mathematics, Rocky Mountain J. Math. Volume 11, Number 4 (1981), 511-552 (doi:10.1216/RMJ-1981-11-4-511)

• E. V. Chuprunov, T. S. Kuntsevich, $n$-Dimensional space groups and regular point systems, Comput. Math. Applic. Vol. 16, No. 5-8, pp. 537-543, 1988 (doi:10.1016/0898-1221(88)90243-X)

• D. Weigel, T. Phan and R. Veysseyre, 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 (doi:10.1107/S0108767387099367)

• Daniel Freed, Gregory Moore, Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)

• Alejandro Tolcachier, Holonomy groups of compact flat solvmanifolds (arXiv:1907.02021)

• GAP package, The Crystallographic Groups Catalog (web)