Contents

group theory

# Contents

## Idea

Crystallographic groups (also: “space groups”) are symmetry groups of crystals.

The use of crystallographic groups for the study of crystals (e.g. Hilton 1903, Engel 1986) is much in the spirit of Klein geometry/the Erlanger program (see for instance Weyl 1938; Grünbaum & Shephard 2010).

Concretely, the quotient space/quotient orbifold of the space of wave vectors/momenta in a crystal lattice by the (dual) crystallographic group is the Brillouin torus(-orbifold), in terms of which most of condensed matter theory is formulated (see for instance the electron energy bands, the valence bundle, and the K-theory classification of topological phases).

## Definition

A crystallographic group or space group in dimension $n$ is a subgroup of the corresponding Euclidean group, hence of the isometry group of Euclidean space $\mathbb{R}^n$, that contains a (maximal) 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 discrete 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:

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

This transparent description of crystallographic groups is essentially the content of Bieberbach's first theorem (following Bieberbach 1910, see Farkas 1981, Thm. 14, Charlap 1986, Thm. I 3.1, Engel 1986, Thm. 3.1 for modern accounts and find a concise statement in Tolcachier 19, Thm. 2.3, also implicitly in Freed-Moore 13, (0.2)). Historically, the earlier definition of “crystallographic group” (eg. Farkas 81, Sec. 3) just required that it be any discrete subgroup of $Iso(E)$ such that the resulting quotient topological group be compact. Bieberbach’s first theorem (or the statement now known under this name) says that this already implies 1. that the translation subgroup is a full lattice, and 2. that the point group is finite.

If the crystallographic short exact sequence on the left of (1) splits, hence if the space group $S \simeq G \ltimes N$ is the semidirect product of the point group with the translational lattice, then $S$ is called a symmorphic space group.

The conjugacy class of a crystallographic group (under conjugation by O(n)) represents the Bravais lattice of the corresponding crystal lattice.

## 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 Schoenflies 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. . 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  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 .

The n-dimensional mathematical crystallography is still in progess. Ryshkov  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 . 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.

Review:

• Harold Hilton, Mathematical crystallography and the theory of groups of movements, Oxford Clarendon Press (1903) $[$web$]$

• Peter Engel, Geometric Crystallography – An Axiomatic Introduction to Crystallography, D. Reidel Publishing (1986) $[$doi:10.1007/978-94-009-4760-3$]$

• Willard Miller, Chapter 2 “The Crystallographic Groups” in : Symmetry Groups and Their Applications, Pure and Applied Mathematics 50 (1972) 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, Geometriae Dedicata 209 (2020) 95–117 $[$arXiv:1907.02021, doi:10.1007/s10711-020-00524-8$]$

• GAP package, The Crystallographic Groups Catalog (web)

Bieberbach’s original articles:

• Ludwig Bieberbach, Über die Bewegungsgruppen des $n$ dimensionalen Euklidischen Raumes mit einem endlichen Fundamentalbereich, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1910) 75-84 $[$dml:58754$]$

• Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Erste Abhandlung), Mathematische Annalen 70 (1911) 297–336 $[$doi:10.1007/BF01564500$]$

• Ludwig Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (ZweiteAbhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Mathematische Annalen 72 (1912) 400-412 $[$doi:10.1007/BF01456724$]$

A monograph on the topic: