International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 1.3, pp. 28-31

## Section 1.3.3. The structure of space groups

B. Souvigniera*

aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

### 1.3.3. The structure of space groups

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#### 1.3.3.1. Point groups of space groups

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The multiplication rule for symmetry operations shows that the mapping which assigns a space-group operation to its linear part is actually a group homomorphism, because the first component of the combined operation is simply the product of the linear parts of the two operations. As a consequence, the linear parts of a space group form a group themselves, which is called the point group of . The kernel of the homomorphism Π consists precisely of the translations , and since kernels of homomorphisms are always normal subgroups (cf. Section 1.1.6 ), the translation subgroup forms a normal subgroup of . According to the homomorphism theorem (see Section 1.1.6 ), the point group is isomorphic to the factor group .

#### Definition

The point group of a space group is the group of linear parts of operations occurring in . It is isomorphic to the factor group of by the translation subgroup .

When is considered with respect to a coordinate system, the operations of are simply 3 × 3 matrices.

The point group plays an important role in the analysis of the macroscopic properties of crystals: it describes the symmetry of the set of face normals and can thus be directly observed. It is usually obtained from the diffraction record of the crystal, where adding the information about the translation subgroup explains the sharpness of the Bragg peaks in the diffraction pattern.

Although we have already deduced that the translation subgroup of a space group forms a normal subgroup in because it is the kernel of the homomorphism mapping each operation to its linear part, it is worth investigating this fact by an explicit computation. Let be a translation in and an arbitrary operation in , then one has which is again a translation in , namely by . This little computation shows an important property of the translation subgroup with respect to the point group, namely that every vector from the translation lattice is mapped again to a lattice vector by each operation of the point group of .

Proposition. Let be a space group with point group and translation subgroup and let be the lattice of translations in . Then acts on the lattice , i.e. for every and one has .

A point group that acts on a lattice is a subgroup of the full group of symmetries of the lattice, obtained as the group of orthogonal mappings that map the lattice to itself. With respect to a primitive basis, the group of symmetries of a lattice consists of all integral basis transformations that fix the metric tensor of the lattice.

#### Definition

Let be a three-dimensional lattice with metric tensor with respect to a primitive basis .

 (i) An automorphism of is an isometry mapping to itself. Written with respect to the basis , an automorphism of is an integral basis transformation fixing the metric tensor of , i.e. it is an integral matrix with . (ii) The group of all automorphisms of is called the automorphism group or Bravais group of . Note that acts on the coordinate columns of , which are simply columns with integral coordinates.

Since the isometries in the Bravais group of a lattice preserve distances, the possible images of the vectors in a basis are vectors of the same lengths as the basis vectors. But due to its discreteness, a lattice contains only finitely many lattice vectors up to a given length. This means that a lattice automorphism can only permute the finitely many vectors up to the maximum length of a basis vector. Thus, there can only be finitely many automorphisms of a lattice. This argument proves the following important fact:

Theorem. The Bravais group of a lattice is finite. As a consequence, point groups of space groups are finite groups.

As subgroups of the Bravais group of a lattice, point groups can be realized as integral matrix groups when written with respect to a primitive basis. For a centred lattice, it is possible that the Bravais group of a lattice contains non-integral matrices, because the centring vector is a column with non-integral entries. However, in dimensions two and three the conventional bases are chosen such that the Bravais groups of all lattices are integral when written with respect to a conventional basis.

Information on the Bravais groups of the primitive lattices in two- and three-dimensional space is displayed in Tables 1.3.3.1 and 1.3.3.2. The columns of the tables contain the names of the lattices, the metric tensor with respect to the conventional basis (with only the upper half given, the lower half following by the symmetry of the metric tensor), the Hermann–Mauguin symbol for the type of the Bravais group and generators of the Bravais group (given in the shorthand notation introduced in Section 1.2.2.1 and the corresponding Seitz symbols discussed in Section 1.4.2.2 ).

 Table 1.3.3.1| top | pdf | Automorphism groups of two-dimensional primitive lattices
LatticeMetric tensorBravais group
Hermann–Mauguin symbolGenerators
Oblique 2 2:
Rectangular 2mm 2:
m10:
Square 4mm 4+:
m10:
Hexagonal 6mm 6+:
m21:
 Table 1.3.3.2| top | pdf | Automorphism groups of three-dimensional primitive lattices
LatticeMetric tensor Bravais group
Hermann–Mauguin symbolGenerators
Triclinic
Monoclinic 2/m

Orthorhombic mmm

Tetragonal 4/mmm

Hexagonal 6/mmm

Rhombohedral

Cubic

The finiteness and integrality of the point groups has important consequences. For example, it implies the crystallographic restriction that rotations in space groups of two- and three-dimensional space can only have orders 1, 2, 3, 4 or 6. On the one hand, an integral matrix clearly has an integral trace.1 But a matrix with the property that can be diagonalized over the complex numbers and the diagonal entries have to be kth roots of unity, i.e. powers of . Since diagonalization does not change the trace, the sum of these kth roots of unity still has to be an integer and in particular these roots of unity have to occur in complex conjugate pairs. In dimension 2 this means that the two diagonal entries are complex conjugate and the only possible ways to obtain an integral trace are , , , and . In dimension 3 the third diagonal entry does not have a complex conjugate partner, and therefore has to be . Thus the possible orders in dimension 3 are the same as in dimension 2.

A much stronger result was obtained by H. Minkowski (1887). He gave an explicit bound for the maximal power of a prime p which can divide the order of an n-dimensional finite integral matrix group. In dimension 2 this theorem implies that the orders of the point groups divide 24 and in dimension 3 the orders of the point groups divide 48. The Bravais groups 4mm (of order 8) and 6mm (of order 12) of the square and hexagonal lattices in dimension 2 and the Bravais group (of order 48) of the cubic lattice in dimension 3 show that Minkowski's result is the best possible in these dimensions.

#### 1.3.3.2. Coset decomposition with respect to the translation subgroup

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The translation subgroup of a space group can be used to distribute the operations of into different classes by grouping together all operations that differ only by a translation. This results in the decomposition of into cosets with respect to (see Section 1.1.4 for details of cosets).

#### Definition

Let be a space group with translation subgroup .

 (i) The right coset of an operation with respect to is the set . Analogously, the set is called the left coset of with respect to . (ii) A set of operations in is called a system of coset representatives relative to if every operation in is contained in exactly one coset . (iii) Writing as the disjoint union is called the coset decomposition of relative to .

If the translation subgroup is a subgroup of index [i] in , a set of coset representatives for relative to consists of [i] operations , where is assumed to be the identity element of . The cosets of relative to can be imagined as columns of an infinite array with [i] columns, labelled by the coset representatives, as displayed in Table 1.3.3.3.

 Table 1.3.3.3| top | pdf | Right-coset decomposition of relative to

Remark: We can assume some enumeration of the operations in because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in are simply columns with integral components . A straightforward enumeration of these columns would start with

Writing out the matrix–column pairs, the coset consists of the operations of the form with running over the lattice translations of . This means that the operations of a coset with respect to the translation subgroup all have the same linear part, which is also evident from a listing of the cosets as columns of an infinite array, as in the example above.

#### Proposition

Let and be two operations of a space group with translation subgroup .

 (1) If , then the cosets and are disjoint, i.e. their intersection is empty. (2) If , then the cosets and are equal, because has linear part and is thus an operation contained in .

The one-to-one correspondence between the point-group operations and the cosets relative to explicitly displays the isomorphism between the point group of and the factor group . This correspondence is also exploited in the listing of the general-position coordinates. What is given there are the coordinate triplets for coset representatives of relative to , which correspond to the first row of the array in Table 1.3.3.3. As just explained, the other operations in can be obtained from these coset representatives by adding a lattice translation to the translational part.

Furthermore, the correspondence between the point group and the coset decomposition relative to makes it easy to find a system of coset representatives of relative to . What is required is that the linear parts of the are precisely the operations in the point group of . If are the different operations in the point group of , then a system of coset representatives is obtained by choosing for every linear part a translation part such that is an operation in .

It is customary to choose the translation parts of the coset representatives such that their coordinates lie between 0 and 1, excluding 1. In particular, if the translation part of a coset representative is a lattice vector, it is usually chosen as the zero vector .

Note that due to the fact that is a normal subgroup of , a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.

#### 1.3.3.3. Symmorphic and non-symmorphic space groups

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If a coset with respect to the translation subgroup contains an operation of the form with a vector in the translation lattice, it is clear that the same coset also contains the operation with trivial translation part. On the other hand, if a coset does not contain an operation of the form , this may be caused by an inappropriate choice of origin. For example, the operation is turned into the inversion by moving the origin to (cf. Section 1.5.1.1 for a detailed treatment of origin-shift transformations).

Depending on the actual space group , it may or may not be possible to choose the origin such that every coset with respect to contains an operation of the form .

#### Definition

Let be a space group with translation subgroup . If it is possible to choose the coordinate system such that every coset of with respect to contains an operation with trivial translation part, is called a symmorphic space group, otherwise is called a non-symmorphic space group.

One sees that the operations with trivial translation part form a subgroup of which is isomorphic to a subgroup of the point group . This subgroup is the group of operations in that fix the origin and is called the site-symmetry group of the origin (site-symmetry groups are discussed in detail in Section 1.4.4 ). It is the distinctive property of symmorphic space groups that they contain a subgroup which is isomorphic to the full point group. This may in fact be seen as an alternative definition for symmorphic space groups.

Proposition. A space group with point group is symmorphic if and only if it contains a subgroup isomorphic to . For a non-symmorphic space group , every finite subgroup of is isomorphic to a proper subgroup of the point group.

Note that every finite subgroup of a space group is a subgroup of the site-symmetry group for some point, because finite groups cannot contain translations. Therefore, a symmorphic space group is characterized by the fact that it contains a site-symmetry group isomorphic to its point group, whereas in non-symmorphic space groups all site-symmetry groups have orders strictly smaller than the order of the point group.

Symmorphic space groups can easily be constructed by choosing a lattice and a point group which acts on . Then is a space group in which the coset representatives can be chosen as .

Non-symmorphic space groups can also be constructed from a lattice and a point group . What is required is a system of coset representatives with respect to and these are obtained by choosing for each operation a translation part . Owing to the translations, it is sufficient to consider vectors with components between 0 and 1. However, the translation parts cannot be chosen arbitrarily, because for a point-group operation of order k, the operation has to be a translation with . Working this out, this imposes the restriction that Once translation parts are found that fulfil all these restrictions, one finally has to check whether the space group obtained this way is (by accident) symmorphic, but written with respect to an inappropriate origin. A change of origin by is realized by conjugating the matrix–column pair by the translation (cf. Section 1.5.1 on transformations of the coordinate system) which givesThus, the space group just constructed is symmorphic if there is a vector such that for each of the coset representatives .

The above considerations also show how every space group can be assigned to a symmorphic space group in a canonical way, namely by setting the translation parts of coset representatives with respect to to . This has the effect that screw rotations are turned into rotations and glide reflections into reflections. The Hermann–Mauguin symbol (see Section 1.4.1 for a detailed discussion of Hermann–Mauguin symbols) of the symmorphic space group to which an arbitrary space group is assigned is simply obtained by replacing any screw rotation symbol Nm by the corresponding rotation symbol N and every glide reflection symbol a, b, c, d, e, n by the symbol m for a reflection. A space group is found to be symmorphic if no such replacement is required, i.e. if the Hermann–Mauguin symbol only contains the symbols 1, 2, 3, 4, 6 for rotations, , , , for rotoinversions and m for reflections.

#### Example

The space groups with Hermann-Mauguin symbols P4mm, P4bm, P42cm, P42nm, P4cc, P4nc, P42mc, P42bc are all assigned to the symmorphic space group with Hermann–Mauguin symbol .

### References

Minkowski, H. (1887). Zur Theorie der positiven quadratischen Formen. J. Reine Angew. Math. 101, 196–202.