Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 70-71   | 1 | 2 |

Section Classification of crystallographic groups

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Classification of crystallographic groups

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Let Γ be a crystallographic group, Λ the normal subgroup of its lattice translations, and G the finite factor group [\Gamma/\Lambda]. Then G acts on Λ by conjugation [Section[link](d)[link]] and this action, being a mapping of a lattice into itself, is representable by matrices with integer entries.

The classification of crystallographic groups proceeds from this observation in the following three steps:

  • Step 1: find all possible finite abstract groups G which can be represented by [3 \times 3] integer matrices.

  • Step 2: for each such G find all its inequivalent representations by [3 \times 3] integer matrices, equivalence being defined by a change of primitive lattice basis (i.e. conjugation by a [3 \times 3] integer matrix with determinant ±1).

  • Step 3: for each G and each equivalence class of integral representations of G, find all inequivalent extensions of the action of G from Λ to [T(3)], equivalence being defined by an affine coordinate change [i.e. conjugation by an element of [A(3)]].

Step 1[link] leads to the following groups, listed in association with the crystal system to which they later give rise:[\matrix{{\bb Z}/2{\bb Z}\hfill &\hbox{monoclinic}\hfill \cr {\bb Z}/2{\bb Z} \oplus {\bb Z}/2{\bb Z}\hfill & \hbox{orthorhombic}\hfill \cr{\bb Z}/3{\bb Z}, ({\bb Z}/3{\bb Z})\, \triangleright\kern-4pt \lt \{\alpha\}\hfill &\hbox{trigonal}\hfill \cr {\bb Z}/4{\bb Z}, ({\bb Z}/4{\bb Z}) \, \triangleright\kern-4pt \lt \{\alpha\}\hfill &\hbox{tetragonal}\hfill \cr {\bb Z}/6{\bb Z}, ({\bb Z}/6{\bb Z}) \, \triangleright\kern-4pt \lt \{\alpha\}\hfill &\hbox{hexagonal}\hfill \cr ({\bb Z}/2{\bb Z} \oplus {\bb Z}/2{\bb Z}) \, \triangleright\kern-4pt \lt \{S_{3}\}\hfill &\hbox{cubic}\hfill}]and the extension of these groups by a centre of inversion. In this list [\triangleright\kern-2pt \lt] denotes a semi-direct product [Section[link](d)[link]], α denotes the automorphism [g \,\longmapsto\, g^{-1}], and [S_{3}] (the group of permutations on three letters) operates by permuting the copies of [{\bb Z}/2{\bb Z}] (using the subgroup [A_{3}] of cyclic permutations gives the tetrahedral subsystem).

Step 2[link] leads to a list of 73 equivalence classes called arithmetic classes of representations [g \,\longmapsto\, {\bf R}_{g}], where [{\bf R}_{g}] is a [3 \times 3] integer matrix, with [{\bf R}_{g_{1} g_{2}} = {\bf R}_{g_{1}} {\bf R}_{g_{2}}] and [{\bf R}_{e} = {\bf I}_{3}]. This enumeration is more familiar if equivalence is relaxed so as to allow conjugation by rational [3 \times 3] matrices with determinant ± 1: this leads to the 32 crystal classes. The difference between an arithmetic class and its rational class resides in the choice of a lattice mode [(P,\ A/B/C,\ I,\ F \hbox { or } R)]. Arithmetic classes always refer to a primitive lattice, but may use inequivalent integral representations for a given geometric symmetry element; while crystallographers prefer to change over to a nonprimitive lattice, if necessary, in order to preserve the same integral representation for a given geometric symmetry element. The matrices P and [{\bf Q} = {\bf P}^{-1}] describing the changes of basis between primitive and centred lattices are listed in Table[link] and illustrated in Figs.[link] to[link] , pp. 80–85, of Volume A of International Tables (Arnold, 2005[link]).

Step 3[link] gives rise to a system of congruences for the systems of nonprimitive translations [\{{\bf t}_{g}\}_{g \in G}] which may be associated to the matrices [\{{\bf R}_{g}\}_{g \in G}] of a given arithmetic class, namely:[{\bf t}_{g_{1}g_{2}} \equiv {\bf R}_{g_{1}} {\bf t}_{g_{2}} + {\bf t}_{g_{1}} \hbox{ mod } \Lambda,]first derived by Frobenius (1911)[link]. If equivalence under the action of [A(3)] is taken into account, 219 classes are found. If equivalence is defined with respect to the action of the subgroup [A^{+}(3)] of [A(3)] consisting only of transformations with determinant +1, then 230 classes called space-group types are obtained. In particular, associating to each of the 73 arithmetic classes a trivial set of nonprimitive translations [({\bf t}_{g} = {\bf 0} \hbox { for all } g \in G)] yields the 73 symmorphic space groups. This third step may also be treated as an abstract problem concerning group extensions, using cohomological methods [Ascher & Janner (1965)[link]; see Janssen (1973)[link] for a summary]; the connection with Frobenius's approach, as generalized by Zassenhaus (1948)[link], is examined in Ascher & Janner (1968)[link].

The finiteness of the number of space-group types in dimension 3 was shown by Bieberbach (1912[link]) to be the case in arbitrary dimension. The reader interested in N-dimensional space-group theory for [N \,\gt \,3] may consult Brown (1969)[link], Brown et al. (1978)[link], Schwarzenberger (1980[link]) and Engel (1986)[link]. The standard reference for integral representation theory is Curtis & Reiner (1962)[link].

All three-dimensional space groups G have the property of being solvable, i.e. that there exists a chain of subgroups[G = G_{r} \,\gt \,G_{r-1}\, \gt \,\ldots \,\gt \,G_{1} \,\gt\, G_{0} = \{e\},]where each [G_{i-1}] is a normal subgroup of [G_{1}] and the factor group [G_{i}/G_{i-1}] is a cyclic group of some order [m_{i}] [(1 \leq i \leq r)]. This property may be established by inspection, or deduced from a famous theorem of Burnside [see Burnside (1911[link]), pp. 322–323] according to which any group G such that [|G| = p^{\alpha} q^{\beta}], with p and q distinct primes, is solvable; in the case at hand, [p = 2] and [q = 3]. The whole classification of 3D space groups can be performed swiftly by a judicious use of the solvability property (L. Auslander, personal communication).

Solvability facilitates the indexing of elements of G in terms of generators and relations (Coxeter & Moser, 1972[link]; Magnus et al., 1976[link]) for the purpose of calculation. By definition of solvability, elements [g_{1}, g_{2}, \ldots, g_{r}] may be chosen in such a way that the cyclic factor group [G_{i}/G_{i-1}] is generated by the coset [g_{i}G_{i-1}]. The set [\{g_{1}, g_{2}, \ldots, g_{r}\}] is then a system of generators for G such that the defining relations [see Brown et al. (1978[link]), pp. 26–27] have the particularly simple form[\eqalign{g_{1}^{m_{1}} &= e, \cr g_{i}^{m_{i}} &= g_{i-1}^{a(i, \, i-1)} g_{i-2}^{a(i, \, i-2)} \ldots g_{1}^{a(i, \, 1)}\phantom{1,2,}\quad \hbox{for } 2 \leq i \leq r, \cr g_{i}^{-1} g_{j}^{-1} g_{i}g_{j} &= g_{j-1}^{b(i, \, j, \, j-1)} g_{j-2}^{b(i, \, j, \, j-2)} \ldots g_{1}^{b(i, \, j, \, 1)}\quad \,\hbox{for } 1 \leq i \,\lt\, j \leq r,}]with [0 \leq a(i, h) \,\lt \,m_{h}] and [0 \leq b(i, j, h) \,\lt\, m_{h}]. Each element g of G may then be obtained uniquely as an `ordered word':[g = g_{r}^{k_{r}} g_{r-1}^{k_{r-1}} \ldots g_{1}^{k_{1}},]with [0 \leq k_{i} \,\lt\, m_{i} \hbox{ for all } i = 1, \ldots, r], using the algorithm of Jürgensen (1970)[link]. Such generating sets and defining relations are tabulated in Brown et al. (1978[link], pp. 61–76). An alternative list is given in Janssen (1973[link], Table 4.3, pp. 121–123, and Appendix D, pp. 262–271).


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Ascher, E. & Janner, A. (1968). Algebraic aspects of crystallography. II. Non-primitive translations in space groups. Commun. Math. Phys. 11, 138–167.
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Zassenhaus, H. (1948). Über eine Algorithmus zur Bestimmung der Raumgruppen. Commun. Helv. Math. 21, 117–141.

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