Chapter 1.2. General introduction to the subgroups of space groups

A crystal is a finite block of an infinite periodic array of atoms in physical space. The infinite periodic array is called the crystal pattern. The finite block is called the macroscopic crystal.

Periodicity implies that there are translations which map the crystal pattern onto itself. Geometric mappings have the property that for each point P of the space, and thus of the object, there is a uniquely determined point , the image point. The mapping is reversible if each image point is the image of one point P only.

Translations belong to a special category of mappings which leave all distances in the space invariant (and thus within an object and between objects in the space). Furthermore, a mapping of an object onto itself (German: Deckoperation) is the basis of the concept of geometric symmetry. This is expressed by the following two definitions.

An isometry is a special kind of affine mapping. In an affine mapping, parallel lines are mapped onto parallel lines; lengths and angles may be distorted but quotients of lengths on the same line are preserved. In Section 1.2.2.3, the description of affine mappings is discussed, because this type of description also applies to isometries. Affine mappings are important for the classification of crystallographic symmetries, cf. Section 1.2.5.2.

Instead of `maps the object onto itself', one frequently says `leaves the object invariant (as a whole)'. This does not mean that each point of the object is mapped onto itself; rather, the object is mapped in such a way that an observer cannot distinguish the states of the object before and after the mapping.

The symmetry operations of a macroscopic crystal are also crystallographic symmetry operations, but they belong to another kind of mapping which will be discussed in Section 1.2.5.4.

There are different types of isometries which may be crystallographic symmetry operations. These types are described and discussed in many textbooks of crystallography and in mathematical, physical and chemical textbooks. They are listed here without further treatment. Fixed points are very important for the characterization of isometries.

The set of all fixed points of an isometry may be the whole space, a plane in the space, a straight line, a point, or the set may be empty (no fixed point).

The following kinds of isometries exist:

Symmetry operations of crystal patterns may belong to any of these isometries. The set of all symmetry operations of a crystal pattern has the following properties: performing two (and thus more) symmetry operations one after the other results in another symmetry operation. Moreover, there is the identity operation in this set, i.e. an operation that leaves every point of the space and thus of the crystal pattern fixed. Finally, for any symmetry operation of an object there is an `inverse' symmetry operation by which its effect is reversed. These properties are necessary for the application of group theory, cf. Section 1.2.3.

To describe mappings analytically, one introduces a coordinate system , consisting of three linearly independent (i.e. not coplanar) basis vectors (or ) and an origin O. For the plane (two-dimensional space) an origin and two linearly independent (i.e. not parallel) basis vectors (or ) are chosen. Referred to this coordinate system, each point P can be described by three (or two for the plane) coordinates (or ). An object, e.g. a crystal, can now be described by a continuous or discontinuous function of the coordinates such as the electron density or the coordinates of the centres of the atoms. A mapping can be regarded as an instruction of how to calculate the coordinates of the image point from the coordinates of the original point X.

In contrast to the practice in physics and chemistry, a non-Cartesian coordinate system is usually chosen in crystallography. The primary aim of the choice of the crystallographic coordinate system is to describe the crystal pattern and its set of all symmetry operations in a simple way. This aim holds in particular for the infinitely many symmetry translations of the crystal pattern, which form its translation group. Secondary to this aim are equality of the lengths of, and right angles between, the basis vectors.

The vector t belonging to the translation t is called a translation vector or a lattice vector. The set of all translation vectors of the crystal pattern is called its vector lattice L. Both the translation group and the vector lattice are useful tools for describing the periodicity of the crystals.

For the description of a vector lattice several kinds of bases are in use. Orthonormal bases are not the most convenient, because the coefficients of the lattice vectors may then be any real number. The coefficients of the lattice vectors are more transparent if the basis vectors themselves are lattice vectors.

Referred to a lattice basis, each lattice vector is a linear combination of the basis vectors with rational coefficients. One can even select special bases such that the coefficients of all lattice vectors are integers.

A fundamental feature of vector lattices is that for any lattice in a dimension greater than one an infinite number of primitive bases exists. With certain rules, the choice of a primitive basis can be made unique (reduced bases). In practice, however, the conventional bases are not always primitive; the choice of a conventional basis is determined by the matrix parts of the symmetry operations, cf. Section 1.2.5.1.

The instruction for the calculation of the coordinates of from the coordinates of X is simple for an affine mapping and thus for an isometry. The equations are where the coefficients and are constant. These equations can be written using the matrix formalism: This matrix equation is usually abbreviated by where

In equations (1.2.2.1) and (1.2.2.3), the coordinates are mixed with the quantities describing the mapping, designated by the letters and or W and w. Therefore, one prefers to write equation (1.2.2.3) in the form The symbols () and {} which describe the mapping referred to the chosen coordinate system are called the matrix–column pair and the Seitz symbol.

The formulae for the combination of affine mappings and for the inverse of an affine mapping (regular matrix W) are obtained by From , it follows that or

Using matrix–column pairs, this reads and or

One finds from equations (1.2.2.5) and (1.2.2.6) that the linear parts of the matrix–column pairs transform as one would expect:

These relations will be used in Section 1.2.5.4.

For the column parts, equations (1.2.2.5) and (1.2.2.6) are less convenient:

Because of the inconvenience of these relations, it is often preferable to use `augmented' matrices, by which one can describe the combination of affine mappings and the inverse mapping by the equations of the usual matrix multiplication. These matrices are introduced in the next section.

It is natural to combine the matrix part and the column part describing an affine mapping to form a matrix, but such matrices cannot be multiplied by the usual matrix multiplication and cannot be inverted. However, if one supplements the matrix by a fourth row `0 0 0 1', one obtains a square matrix which can be combined with the analogous matrices of other mappings and can be inverted. These matrices are called augmented matrices and are designated by open-face letters in this volume:

In order to write equation (1.2.2.3) as with the augmented matrices , the columns and x also have to be extended to the augmented columns and . Equations (1.2.2.5) and (1.2.2.6) then become

The vertical and horizontal lines in the matrix have no mathematical meaning. They are simply a convenience for separating the matrix part from the column part and from the row `0 0 0 1', and could be omitted.

Augmented matrices are very useful when writing down general formulae which then become more transparent and more elegant. However, the matrix–column pair formalism is, in general, advantageous for practical calculations.

For the augmented columns of vector coefficients, see Section 1.2.2.6.

Isometries are special affine mappings, as in Definition 1.2.2.1.1. The matrix W of an isometry has to fulfil conditions which depend on the coordinate basis. These conditions are:

Algorithms for the determination of the kind of isometry from a given matrix–column pair and for the determination of the matrix–column pair for a given isometry can be found in IT A, Part 11 or in Hahn & Wondratschek (1994).

In crystallography, vectors and their coefficients as well as points and their coordinates are used for the description of crystal structures. Vectors represent translation shifts, distance and Patterson vectors, reciprocal-lattice vectors etc. With respect to a given basis a vector has three coefficients. In contrast to the coordinates of a point, these coefficients do not change if the origin of the coordinate system is shifted. In the usual description by columns, the vector coefficients cannot be distinguished from the point coordinates, but in the augmented-column description the difference becomes visible: the vector from the point P to the point Q has the coefficients , , , . Thus, the column of the coefficients of a vector is not augmented by `1' but by `0'. Therefore, when the point P is mapped onto the point by according to equation (1.2.2.3), then the vector is mapped onto the vector by transforming its coefficients by , because the coefficients are multiplied by the number `0' augmenting the column . Indeed, the distance vector is not changed when the whole space is mapped onto itself by a translation.

Remarks :

It is in general advantageous to refer crystallographic objects and their symmetries to the most appropriate coordinate system. The best coordinate system may be different for different steps of the calculations and for different objects which have to be considered simultaneously. Therefore, a change of the origin and/or the basis are frequently necessary when treating crystallographic problems. Here the formulae for the influence of an origin shift and a change of basis on the coordinates, on the matrix–column pairs of mappings and on the vector coefficients are only stated; the equations are derived in detail in IT A Chapters 5.1 and 5.2 , and in Hahn & Wondratschek (1994).

Let a coordinate system be given with a basis and an origin O.¹ Referred to this coordinate system, the column of coordinates of a point P is x; the matrix and column parts describing a symmetry operation are W and w according to equations (1.2.2.1) to (1.2.2.3), and the column of vector coefficients is v, see Section 1.2.2.6. A new coordinate system may be introduced with the basis and the origin . Referred to the new coordinate system, the column of coordinates of the point P is , the symmetry operation is described by and and the column of vector coefficients is .

Let be the column of coefficients for the vector from the old origin O to the new origin and letbe the matrix of a basis change, i.e. the matrix that relates the new basis to the old basis according to Then the following equations hold: For the columns of vector coefficients v and , the following holds: i.e. an origin shift does not change the vector coefficients.

These equations read in the augmented-matrix formalism For the difference in the transformation behaviour of point coordinates and vector coefficients, see the remarks at the end of Section 1.2.2.6. A vector v can be regarded as a translation vector; its translation is then described by (I, v), i.e. . It can be shown using equation (1.2.2.13) that the translation and thus the translation vector are not changed under an origin shift, (), because holds. Moreover, under a general coordinate transformation the origin shift is not effective: in equation (1.2.2.13) only remains because of the equality .

The geometric symmetry of any object is described by a group . The symmetry operations are the group elements, and the set of all symmetry operations fulfils the group postulates. [A `symmetry element' in crystallography is not a group element of a symmetry group but is a combination of a geometric object with that set of symmetry operations which leave the geometric object invariant, e.g. an axis with its threefold rotations or a plane with its glide reflections etc., cf. Flack et al. (2000).] Groups will be designated by upper-case calligraphic script letters , etc. Group elements are represented by lower-case sans serif letters etc.

The result of the composition of two elements will be called the product of and and will be written . The first operation is the right factor because the point coordinates or vector coefficients are written as columns on which the matrices of the symmetry operations are applied from the left side.

The law of composition in the group is the successive application of the symmetry operations.

The group postulates are shown to hold for symmetry groups:

The number of elements of a group is called its order . The order of a group may be finite, e.g. 24 for the symmetry operations of a regular tetrahedron, or infinite, e.g. for any space group because of its infinite set of translations. If the relation is fulfilled for all pairs of elements of a group , then is called a commutative or an Abelian group.

For groups of higher order, it is usually inappropriate and for groups of infinite order it is impossible to list all elements of a group. The following definition nearly always reduces the set of group elements to be listed explicitly to a small set.

A group is cyclic if it consists of the unit element and all powers of one element :

If there is an integer number with and n is the smallest number with this property, then the group has the finite order n. Let with be the inverse element of where n is the order of . Because with , the elements of a cyclic group of finite order can all be written as positive powers of the generator . Otherwise, if such an integer n does not exist, the group is of infinite order and the positive powers are different from the negative ones .

In the same way, from any element its cyclic group can be generated even if is not cyclic itself. The order of this group is called the order of the element .

A finite group of small order may be conveniently visualized by its multiplication table, group table or Cayley table. An example is shown in Table 1.2.3.1.

Table 1.2.3.1| top | pdf | Multiplication table of a group The group elements are listed at the top of the table and in the same sequence on the left-hand side; the unit element `' is listed first. The table is thus a square array. The product of any pair of elements is listed at the intersection of the kth row and the jth column. It can be shown that each group element is listed exactly once in each row and once in each column of the table. In the row of an element , the unit element appears in the column of . If , i.e. , appears on the main diagonal. The multiplication table of an Abelian group is symmetric about the main diagonal.

The multiplication tables can be used to define one of the most important relations between two groups, the isomorphism of groups. This can be done by comparing the multiplication tables of the two groups.

Multiplication tables are useful only for groups of small order. To define `isomorphism' for arbitrary groups, one can formulate the relations expressed by the multiplication tables in a more abstract way.

The `same multiplication table' for the groups and means that there is a reversible mapping of the elements and such that holds for any pair of indices j and k. In words:

Isomorphic groups have the same order. By isomorphism the set of all groups is classified into isomorphism types or isomorphism classes of groups. Such a class is often called an abstract group.

The isomorphism between the space groups and the corresponding matrix groups makes an analytical treatment of crystallographic symmetry possible. Moreover, the isomorphism of different space groups allows one to classify the infinite number of space groups into a finite number of isomorphism types of space groups, which is one of the bases of crystallography, see Section 1.2.5.

Isomorphism provides a very strong relation between groups: the groups are identical in their group-theoretical properties. One can weaken this relation by omitting the condition of reversibility of the mapping. One then admits that more than one element of the group is mapped onto the same element of . This concept leads to the definition of homomorphism.

The formulation `mapping onto' implies that each element occurs among the images of the elements at least once.²

The very important concept of homomorphism is discussed further in Lemma 1.2.4.4.3. The crystallographic point groups are homomorphic images of the space groups, see Section 1.2.5.4.

There may be sets of elements that do not constitute the full group but nevertheless fulfil the group postulates for themselves.

Remarks :

This definition is very important for the tables of this volume, as only maximal subgroups of space groups are listed. If all maximal subgroups are known for any given space group, then any general subgroup can be obtained by a (finite) chain of maximal subgroups between and , see Section 1.2.6.2. Moreover, the relations between a space group and its maximal subgroups are particularly transparent, cf. Lemma 1.2.8.1.3.

Let be a subgroup of of order . Because is a proper subgroup of there must be elements that are not elements of . Let be one of them. Then the set of elements ³ is a subset of elements of with the property that all its elements are different and that the sets and have no element in common. Thus, the set also contains elements of . If there is another element which belongs neither to nor to , one can form another set . All elements of are different and none occurs already in or in . This procedure can be continued until each element belongs to one of these sets. In this way the group can be partitioned, such that each element belongs to exactly one of these sets.

Remarks :

From its definition and from the properties of the coset decomposition mentioned above, one immediately obtains the fundamental theorem of Lagrange (for another formulation, see Chapter 1.5 ):

A special situation exists when the left and right coset decompositions of relative to result in the partition of into the same cosets: Subgroups that fulfil equation (1.2.4.1) are called `normal subgroups' according to the following definition:

The relation always holds for , i.e. subgroups of index 2 are always normal subgroups. The subgroup contains half of the elements of , whereas the other half of the elements forms `the other' coset. This coset must then be the right as well as the left coset.

In a coset decomposition, the set of all elements of the group is partitioned into cosets which form classes in the mathematical sense of the word, i.e. each element of belongs to exactly one coset.

Another equally important partition of the group into classes of elements arises from the following definition:

Remarks :

Not only the individual elements of a group but also the subgroups of can be classified in conjugacy classes.

Remarks :

Equation (1.2.4.1) can be written Using conjugation, Definition 1.2.4.2.3 can be formulated as

For the following definition, the `product of sets of group elements' will be used:

The coset decomposition of a group relative to a normal subgroup has a property which makes it particularly useful for displaying the structure of a group.

Consider the coset decomposition with the cosets and of a group relative to its subgroup . In general the product of two cosets, cf. Definition 1.2.4.4.1, will not be a coset again. However, if and only if is a normal subgroup of , the product of two cosets is always another coset. This means that for the set of all cosets of a normal subgroup there exists a law of composition for which the closure is fulfilled. One can show that the other group postulates are also fulfilled for the cosets and their multiplication if holds: there is a neutral element (which is ), for each coset the coset forms the inverse element and for the coset multiplication the associative law holds.

A factor group is not necessarily isomorphic to a subgroup .

Factor groups are indispensable for an understanding of the homomorphism of one group onto the other. The relations between a group and its homomorphic image are very strong and are expressed by the following lemma:

The most important homomorphism in crystallography is the relation between a space group and its homomorphic image, the point group , where the kernel is the subgroup () of all translations of , cf. Section 1.2.5.4.

The concept of the normalizer of a group in a group is very useful for the considerations of the following sections. The size of the conjugacy class of in is determined by this normalizer.

Let and . Then holds because is a group. If , then for any . If is not a normal subgroup of , there may nevertheless be elements for which holds. We consider the set of all elements that have this property.

Remarks :

The set of all symmetry operations of a three-dimensional crystal pattern, i.e. its symmetry group, is the space group of this crystal pattern. In a plane, the symmetry group of a two-dimensional crystal pattern is its plane group. In the following, the term `space group' alone will be used and the plane groups are included because they are the space groups of two-dimensional space.

A crystal pattern is a periodic array. This means that there are translations among its symmetry operations. The translations of crystals are small (a few angströms to a few hundred angströms) but cannot be arbitrarily short because of the finite size of the particles in crystal structures. One thus defines for any finite integer n:

Condition (2) is justified because crystal structures contain atoms of finite size, and it is necessary to avoid infinitely small translations as elements of space groups. Several fundamental properties would not hold without this condition, such as the existence of a lattice of translation vectors and the restriction to only a few rotation angles.

In this volume, only the dimensions and will be dealt with. However, the space groups (more precisely, the space-group types, cf. Section 1.2.5.3) and other crystallographic items are also known for dimensions and ; the number of the affine space-group types is even known for : 28 927 922 (Plesken & Schulz, 2000).

One of the characteristics of a space group is its translation group. Any space group is an infinite group because the number of its translations is already infinite. The set of all translations of forms the infinite translation subgroup with the composition law of performing one translation after the other, represented by the multiplication of matrix–column pairs. The group is a normal subgroup of of finite index. The vector lattice L, cf. Section 1.2.2.2, forms a group with the composition law of vector addition. This group is isomorphic to the group ().

The matrix–column pairs of the symmetry operations of a space group are mostly referred to the conventional coordinate system. Its basis is chosen as a lattice basis and in such a way that the matrices for the linear parts of the symmetry operations of are particularly simple. The origin is chosen such that as many coset representatives as possible can be selected with their column coefficients to be zero, or such that the origin is situated on a centre of inversion. This means (for details and examples see Section 8.3.1 of IT A):

It is obviously impossible to list all elements of an infinite group individually. One could define the space group by a set of generators, because the number of necessary generators for any space group is finite: theoretically, up to six generators might be necessary but in practice up to ten generators are chosen for a space group. In IT A and in this volume, the set of the conventional generators is listed in the block `Generators selected'. The unit element is taken as the first generator; the generating translations follow and the generation is completed with the generators of the non-translation symmetry operations. The rules for the choice of the conventional generators are described in IT A, Section 8.3.5 .

The description by generators is particularly important for this volume because many of the maximal subgroups in Chapters 2.2 and 2.3 are listed by their generators. These generators are chosen such that the generation of the general position can follow a composition series, cf. Ledermann (1976). This procedure allows the generation by a short program or even by hand. For details see IT A, Section 8.3.5 ; in Table 8.3.5.2 of IT A an example for the generation of a space group along these lines is displayed.

There are four ways to describe a space group in IT A:

There are an infinite number of space groups because there are an infinite number of known or conceivable crystals and crystal patterns. Indeed, because the lattice parameters depend on temperature and pressure, so do the lattice translations and the space group of a crystal. There is great interest in getting an overview of this vast number of space groups. To achieve this goal, one first characterizes the space groups by their group-theoretical properties and classifies them into space-group types where the space groups of each type have certain properties in common. To get a better overview, one then classifies the space-group types such that related types belong to the same `superclass'. This classification is done in two ways (cf. Sections 1.2.5.4 and 1.2.5.5):

In reality, the tables in Chapters 2.2 and 2.3 and the graphs in Chapters 2.4 and 2.5 are tables and graphs for space-group types. The sequence of the space-group types in IT A and thus in this volume is determined by their crystal class, their crystal system and their crystal family. Therefore, these classifications are treated in the next sections. The point groups and the translation groups of the space groups can also be classified in a similar way. Only the classification of the point groups is treated in this chapter. For a more detailed treatment and for the classification of the lattices, the reader is referred to Chapter 1.5 of this volume, to Part 8 of IT A or to Brown et al. (1978).

We first consider the classification of the space groups into types. A more detailed treatment may be found in Section 8.2.1 of IT A. In practice, a common way is to look for the symmetry of the space group and to compare this symmetry with that of the diagrams in the tables of IT A.

With the exception of some double descriptions,⁵ there is exactly one set of diagrams which displays the symmetry of , and belongs to that space-group type which is described in this set. From those diagrams the Hermann–Mauguin symbol, abbreviated as HM symbol, the Schoenflies symbol and the space-group number are taken.

A rigorous definition is:

This definition refers to a rather abstract property which is of great mathematical but less practical value. In crystallography another definition is more appropriate which results in exactly the same space-group types as are obtained by isomorphism. It starts from the description of the symmetry operations of a space group by matrix–column pairs or, as will be formulated here, by augmented matrices. For this one refers each of the space groups to one of its lattice bases.

In this definition the coordinate systems are chosen such that the groups of augmented matrices agree. It is thus possible to describe the symmetry of all space groups of the same type by one (standardized) set of matrix–column pairs, as is done, for example, in the tables of IT A.

In the subgroup tables of Chapters 2.2 and 2.3 it frequently happens that a subgroup of a space group is given by its matrix–column pairs referred to an nonconventional coordinate system. In this case, a transformation of the coordinate system can bring the matrix–column pairs to the standard form by which the space-group type may be determined. In the subgroup tables both the space-group type and the transformation of the coordinate system are listed. One can also use this procedure for the definition of the term `affine space-group type':

The affine space-group types are classes in the mathematical sense of the word, i.e. each space group belongs to exactly one type. The derivation of these types reveals 219 affine space-group types and 17 plane-group types.

In crystallography one usually distinguishes 230 rather than 219 space-group types in a slightly finer subdivision. The difference can best be explained using Definition 1.2.5.3.3. The matrix part P may have a negative determinant. In this case, a right-handed basis is converted into a left-handed one, and right-handed and left-handed screw axes are exchanged. It is a convention in crystallography to always refer the space to a right-handed basis and hence transformations with are not admitted.

The space groups of an enantiomorphic pair belong to different crystallographic space-group types but are isomorphic. As a consequence, in the lists of isomorphic subgroups of the tables of Chapter 2.3 , there may occur subgroups with another conventional HM symbol and another space-group number than that of , cf. Example 1.2.6.2.7. In such a case, and are members of an enantiomorphic pair of space groups and belongs to the space-group type enantiomorphic to that of . There are no enantiomorphic pairs of plane groups.

The space groups are of different complexity. The simplest ones are the symmorphic space groups (not to be confused with `isomorphic' space groups) according to the following definition:

The group is finite and thus leaves a point F fixed. In the standard setting of any symmorphic space group such a point F is chosen as the origin. Thus, the translation parts of the elements consist of zeroes only.

If a space group is symmorphic then all space groups of its type are symmorphic. Therefore, one can speak of `symmorphic space-group types'. Symmorphic space groups can be recognized easily by their HM symbols: they contain an unmodified point-group symbol: rotations, reflections, inversions and rotoinversions but no screw rotations or glide reflections. There are 73 symmorphic space-group types of dimension three and 13 of dimension two; none of them show enantiomorphism.

One frequently speaks of `the 230 space groups' or `the 17 plane groups' and does not distinguish between the terms `space group' and `space-group type'. This is very often possible and is also done in this volume in order to make the explanations less long-winded. However, occasionally the distinction is indispensable in order to avoid serious difficulties of comprehension. For example, the sentence `A space group is a proper subgroup of itself' is incomprehensible, whereas the sentence `A space group and its proper subgroup belong to the same space-group type' makes sense.

If the point coordinates are mapped by an isometry and its matrix–column pair, the vector coefficients are mapped by the linear part, i.e. by the matrix alone, cf. Section 1.2.2.6. Because the number of its elements is infinite, a space group generates from one point an infinite set of symmetry-equivalent points by its matrix–column pairs. Because the number of matrices of the linear parts is finite, the group of matrices generates from one vector a finite set of symmetry-equivalent vectors, e.g. the vectors normal to certain planes of the crystal. These planes determine the morphology of the ideal macroscopic crystal and its cleavage; the centre of the crystal represents the zero vector. When the symmetry of a crystal can only be determined by its macroscopic properties, only the symmetry group of the macroscopic crystal can be found. All its symmetry operations leave at least one point of the crystal fixed, viz its centre of mass. Therefore, this symmetry group was called the point group of the crystal, although its symmetry operations are those of vector space, not of point space. Although misunderstandings are not rare, this name is still used in today's crystallography for historical reasons.⁷

Let a conventional coordinate system be chosen and the elements be represented by the matrix–column pairs , with the representation of the translations by the pairs . Then the composition of with all translations forms an infinite set of symmetry operations which is a right coset of the coset decomposition . From this equation it follows that the elements of the same coset of the decomposition have the same linear part. On the other hand, elements of different cosets have different linear parts if contains all translations of . Thus, each coset can be characterized by its linear part. It can be shown from equations (1.2.2.5) and (1.2.2.6) that the linear parts form a group which is isomorphic to the factor group , i.e. to the group of the cosets.

According to Definition 1.2.5.4.1, the factor group is isomorphic to the point group . This property is exploited in the graphs of translationengleiche subgroups of space groups, cf. Chapter 2.4 and Section 2.1.7.2 .

All point groups in the following sections are crystallographic point groups. The maximum order of a crystallographic point group is 48 in three-dimensional space and 12 in two-dimensional space.

As with space groups, there are also an infinite number of crystallographic point groups which may be classified into a finite number of point-group types. This cannot be done by isomorphism because geometrically different point groups may be isomorphic. For example, point groups consisting of the identity with the inversion or with a twofold rotation or with a reflection through a plane are all isomorphic to the (abstract) group of order 2. As for space groups, the classification may be performed, however, referring the point groups to corresponding vector bases. As translations do not occur among the point-group operations, one may choose any basis for the description of the symmetry operations by matrices. One takes the basis of as given and transforms the basis of to the basis corresponding to that of . This leads to the definition:

Point groups can be classified by Definition 1.2.5.4.2. Further space groups may be classified into `crystal classes of space groups' according to their point groups:

Whether two space groups belong to the same crystal class or not can be worked out from their standard HM symbols: one removes the lattice parts from these symbols as well as the constituents `1' from the symbols of trigonal space groups and replaces all constituents for screw rotations and glide reflections by those for the corresponding pure rotations and reflections. The symbols obtained in this way are those of the corresponding point groups. If they agree, the space groups belong to the same crystal class. The space groups also belong to the same crystal class if the point-group symbols belong to the pair and or to the pair and .

There are 32 classes of three-dimensional crystallographic point groups and 32 crystal classes of space groups, and ten classes of two-dimensional crystallographic point groups and ten crystal classes of plane groups.

The distribution into crystal classes classifies space-group types – and thus space groups – and crystallographic point groups. It does not classify the infinite set of all lattices into a finite number of lattice types, because the same lattice may belong to space groups of different crystal classes. For example, the same lattice may be that of a space group of type (of crystal class 1) and that of a space group of type (of crystal class ).

Nevertheless, there is also a definition of the `point group of a lattice'. Let a vector lattice L of a space group be referred to a lattice basis. Then the linear parts W of the matrix–column pairs (W, w) of form the point group . If (W, w) maps the space group onto itself, then the linear part W maps the (vector) lattice L onto itself. However, there may be additional matrices which also describe symmetry operations of the lattice L. For example, the point group of a space group of type consists of the identity 1 only. However, with any vector , the negative vector also belongs to . Therefore, the lattice is always centrosymmetric and has the inversion as a symmetry operation independent of the symmetry of the space group.

There are seven holohedral crystal classes in the space: , , , , , and . Their lattices are called triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal and cubic, respectively. There are four holohedral crystal classes in the plane: , , and . Their two-dimensional lattices (or nets) are called oblique, rectangular, square and hexagonal, respectively.

The lattices can be classified into lattice types or Bravais types, mostly called Bravais lattices, or into lattice systems (called Bravais systems in editions 1 to 4 of IT A). These classifications are not discussed here because they are not directly relevant to the classification of the space groups. This is because the lattice symmetry is not necessarily typical for the symmetry of its space group but may accidentally be higher. For example, the lattice of a monoclinic crystal may be accidentally orthorhombic (only for certain values of temperature and pressure). In Sections 8.2.5 and 8.2.7 of IT A the `typical lattice symmetry' of a space group is defined.

The example of mentioned above shows that the point group of the lattice may be systematically of higher order than that of its space group. There are obviously point groups and thus space groups that belong to a holohedral crystal class and those that do not. The latter can be assigned to a holohedral crystal class uniquely according to the following definition:⁸

The 32 crystal classes of space groups are classified into seven crystal systems which are called triclinic, monoclinic, ortho­rhombic, tetragonal, trigonal, hexagonal and cubic. There are four crystal systems of plane groups: oblique, rectangular, square and hexagonal. Like the space groups, the crystal classes of point groups are classified into the seven crystal systems of point groups.

Apart from accidental lattice symmetries, the space groups of different crystal systems have lattices of different symmetry. As an exception, the hexagonal primitive lattice occurs in both hexagonal and trigonal space groups as the typical lattice. Therefore, the space groups of the trigonal and the hexagonal crystal systems are more related than space groups from other different crystal systems. Indeed, in different crystallographic schools the term `crystal system' was used for different objects. One sense of the term was the `crystal system' as defined above, while another sense of the old term `crystal system' is now called a `crystal family' according to the following definition [for definitions that are also valid in higher-dimensional spaces, see Brown et al. (1978) or IT A, Chapter 8.2 ]:

The partition of the space groups into crystal families is the most universal one. The space groups and their types, their crystal classes and their crystal systems are classified by the crystal families. Analogously, the crystallographic point groups and their crystal classes and crystal systems are classified by the crystal families of point groups. Lattices, their Bravais types and lattice systems can also be classified into crystal families of lattices; cf. IT A, Chapter 8.2 .

Group–subgroup relations form an essential part of the applications of space-group theory. Let be a space group and a proper subgroup of . All maximal subgroups of any space group are listed in Part 2 of this volume. There are different kinds of subgroups which are defined and described in this section. The tables and graphs of this volume are arranged according to these kinds of subgroups. Moreover, for the different kinds of subgroups different data are listed in the subgroup tables and graphs.

Let and be space groups of the space-group types and . The group–subgroup relation is a relation between the particular space groups and but it can be generalized to the space-group types and . Certainly, not every space group of the type will be a subgroup of every space group of the type . Nevertheless, the relation holds for any space group of and in the following sense: If holds for the pair and , then for any space group of the type a space group of the type exists for which the corresponding relation holds. Conversely, for any space group of the type a space group of the type exists for which the corresponding relation holds. Only this property of the group–subgroup relations made it possible to compile and arrange the tables of this volume so that they are as concise as those of IT A.

`Maximal subgroups' have been introduced by Definition 1.2.4.1.2. The importance of this definition will become apparent in the corollary to Hermann's theorem, cf. Lemma 1.2.8.1.3. In this volume only the maximal subgroups are listed for any plane and any space group. A maximal subgroup of a plane group is a plane group, a maximal subgroup of a space group is a space group. On the other hand, a minimal supergroup of a plane group or of a space group is not necessarily a plane group or a space group, cf. Section 2.1.6 .

If the maximal subgroups are known for each space group, then each non-maximal subgroup of a space group with finite index can in principle be obtained from the data on maximal subgroups. A non-maximal subgroup of finite index [i] is connected with the original group through a chain , where each group is a maximal subgroup of , with the index , . The number k is finite and the relation holds, i.e. the total index [i] is the product of the indices .

According to Hermann (1929), the following types of subgroups of space groups have to be distinguished:

The order of a crystallographic point group of the space group is always finite. Therefore, the number of the subgroups of is also always finite and these subgroups and their relations are displayed in well known graphs, cf. Chapter 2.4 and Section 2.1.7 of this volume. Because of the isomorphism between the point group and the factor group , the subgroup graph for the point group is the same as that for the t-subgroups of , only the labels of the groups are different. For deviations between the point-group graphs and the actual space-group graphs of Chapter 2.4 , cf. Section 2.1.7.2 .

For a klassengleiche subgroup , the cosets of the factor group are smaller than those of . Because () is still infinite, the number of elements of each coset is infinite but the index is finite. The number of k-subgroups of is always infinite.

Isomorphic subgroups are special k-subgroups. The importance of the distinction between k-subgroups in general and isomorphic subgroups in particular stems from the fact that the number of maximal non-isomorphic k-subgroups of any space group is finite, whereas the number of maximal isomorphic subgroups is always infinite, see Section 1.2.8.

If belongs to a pair of enantiomorphic space-group types, then the isomorphic subgroups of may belong to different crystallographic space-group types with different HM symbols and different space-group numbers. In this case, an infinite number of subgroups belong to the crystallographic space-group type of and another infinite number belong to the enantiomorphic space-group type.

In Section 1.2.4.5, the normalizer of a subgroup in the group was defined. The equation holds, i.e. is a normal subgroup of . The normalizer , by its index in , determines the number of subgroups that are conjugate in the group , cf. Remarks (2) and (3) below Definition 1.2.4.5.1.

The group–subgroup relations between space groups become more transparent if one looks at them from a more general point of view. Space groups are part of the general theory of mappings. Particular groups are the affine group of all reversible affine mappings, the Euclidean group of all isometries, the translation group of all translations and the orthogonal group of all orthogonal mappings.

Connected with any particular space group are its group of translations and its point group . In addition, the normalizers of in the affine group and in the Euclidean group are useful. They are listed in Section 15.2.1 of IT A. Although consisting of isometries only, is not necessarily a space group, see the paragraph below Lemma 1.2.7.2.6.

For the group–subgroup pairs the following relations hold:

The subgroup may be a translationengleiche or a klassen­glei­che or a general subgroup of . In any case, the normalizer determines the length of the conjugacy class of , but it is not feasible to list for each group–subgroup pair its normalizer . Indeed, it is only necessary to list for any space group its normalizer in the Euclidean group of all isometries, as is done in IT A, Section 15.2.1 . From such a list the normalizers for the group–subgroup pairs can be obtained easily, because for any chain of space groups , the relations and hold. The normalizer consists consequently of all those isometries of that are also elements of , i.e. that belong to the intersection , cf. the examples of Section 1.2.7.¹⁰

The isomorphism type of the Euclidean normalizer may depend on the lattice parameters of the space group (specialized Euclidean normalizer). For example, if the lattice of the space group of a triclinic crystal is accidentally monoclinic at a certain temperature and pressure or for a certain composition in a continuous solid-solution series, then the Euclidean normalizer of this space group belongs to the space-group types or , otherwise it belongs to . Such a specialized Euclidean normalizer (here or ) may be distinguished from the typical Euclidean normalizer (here ), for which the lattice of is not more symmetric than is required by the symmetry of . The specialized Euclidean normalizers were first listed in the 5th edition of IT A (2005), Section 15.2.1 .

In this section, the group-theoretical aspects of domain (twin) formation (domain structure, transformation twin) from a homogeneous single crystal (phase A, parent phase) to a crystalline phase B (daughter phase, deformed phase) are discussed, where the space group of phase B is a subgroup of the space group of phase A, . This happens, e.g., in a displacive or order–disorder phase transition. In most cases phase B, the domain structure, is inhomogeneous, consisting of homogeneous regions which are called domains, defined below.

Only the basic group-theoretical relations are considered here. A deeper discussion of domain structures and their properties needs methods using representation theory, thermodynamic points of view (Landau theory), lattice dynamics and tensor properties of crystals. Such treatments are beyond the scope of this section. A detailed discussion of them is given by Tolédano et al. (2003) and by Janovec & Přívratská (2003).

In order to make the group-theoretical treatment possible, the parent-clamping approximation, abbreviated PCA, is introduced, by which the lattice parameters of phase A are not allowed to change at and after the transition to phase B, cf. Janovec & Přívratská (2003). Under the assumption of the PCA, two essential conditions hold:

The domain walls, i.e. the boundaries which separate different domains, will not be treated here because their symmetries are layer groups which are two-periodic groups in three-dimensional space and not space groups with three-dimensional periodicity. Layer groups are described in International Tables for Crystallography, Vol. E (2002).

Under these assumptions the domains formed may exhibit different chiralities and polarities of their structures and different spatial orientations of their symmetry elements, but each domain has the same specific energy and the lattice of each domain is part of the lattice of the parent structure A with space group .

In the discussion of domain structures, the following basic concepts are established: domain, domain state, symmetry state, orientation state. These concepts are defined and then applied in different examples of phase transitions in which the group-theoretical procedures and their results are explained.

The second step, the physically realistic situation at a temperature with the removal of the PCA, is only partly considered in this section. The relaxation of the PCA does not change the relations in a non-ferroelastic phase because all crystal regions suffer the same affine deformation. On the other hand, in ferroelastic phases the different spontaneous strains complicate the relations.

In a (continuous) phase transition with the symmetry reduction of the space group to a subgroup , a splitting of the parent phase A into many crystals of the type B is observed. The number of such crystals of B is not limited; they differ in their locations in space, in their orientations, in their shapes and in their space groups which, however, all belong to the same space-group type. In order to describe what happens in such a transition, a few notions are useful. If not explicitly stated, the validity of the PCA is assumed.

If the domains of phase B have been formed from a single crystal of phase A, then relations between the domains exist which are determined by group theory. In particular, the domains belong to a finite (small) number of domain states which have well defined relations to the original crystal A and its space group . In order to describe the relation of B to A, the notion of crystal pattern is used. Any perfect (ideal) crystal is a finite block of the corresponding infinite arrangement, the symmetry of which is a space group which contains translations. Here, this (infinite) periodic object is called a crystal pattern, cf. Section 1.2.2.1.

The number of domain states which are observed after a phase transition is limited and determined by the space groups and . The number of domains which belong to the same domain state is not limited. The diversity of the domains and their shapes is due to mechanical stresses, defects, electrical charges and nucleation phenomena which strongly influence the kinetics of the phase transition.

A trivial domain structure is formed when phase B consists of one domain only, i.e. when it forms a single-domain structure. This is possible, in particular under an external electric field or under external stress. Such a procedure is known as `detwinning'. The corresponding domain state is a single-domain state. For a phase transition of the type considered, there are always several single-domain states which have the same a priori probability of appearing after a phase transition. In reality not all of them will be observed and/or their relative frequencies and sizes will be rather different.

Single-domain states are introduced in theoretical considerations in order to avoid the complications which may be caused by the coexistence of domains with different spontaneous strain in ferroelastic crystals of the structure B if the PCA cannot be assumed. In polydomain structures, the domains would distort or rotate each other a little and thus disturb the simple relations described now. These disturbances do not occur in non-ferroelastic transitions, so for them the simple relations also hold in polydomain structures without the PCA.

If in a group–subgroup relation with index the subgroups belong to more than one conjugacy class, then each conjugacy class corresponds to a separate phase transition , etc. These different phase transitions lead to different low-symmetry structures , have different transition temperatures and different probabilities of happening.

There are more elements of the group than just that map the domain state onto the domain state . The elements of the space group map the domain state of onto itself: , . Therefore, not just the element but all elements of the coset map the domain state of onto the domain state : . This can be expressed in the form:

There is a one-to-one correspondence between the cosets of the decomposition and the possible domain states which may be observed after the transition.

Similarly, all elements map the space group onto by conjugation: .

Note that due to these formulae the i different domain states do not necessarily belong to i different space groups and that different domain states may belong to the same space group, cf. Lemma 1.2.7.2.6 and Example 1.2.7.2.4.

The terms just defined shall be explained in a few examples. By Example 1.2.7.2.4 a translationengleiche transition is displayed; i.e. is a translationengleiche subgroup of . Then the relation between and is essentially reflected by the relation between the point groups and .

Note that here as in many other places of this section one has to distinguish strictly between `space group' as a specimen, e.g. in `space group of a crystal', and `space-group type', which is one of the 230 classes frequently called simply but inexactly `the 230 space groups', see the last paragraph of Section 1.2.5.3.

Domains of the same domain state always belong to the same symmetry state. Domains of different domain states may or may not belong to the same symmetry state. The number of symmetry states is limited and is smaller than or equal to the number of domain states. Moreover, the number of symmetry states is determined by the space groups and .

Let be the normalizer of the space group in the space group . Then with the indices and with . By Lemma 1.2.7.2.3, the number of domain states is determined. For the number of symmetry states the following lemma holds:

For the perovskite transition of Example 1.2.7.2.4, the normalizer can be obtained from the Euclidean normalizer of in Table 15.2.1.4 of IT A which is listed as . This Euclidean normalizer has continuous translations along the z direction (indicated by the lattice part of the HM symbol) and is thus not a space group. However, all additional translations of are not elements of the space group , and is a subgroup of with index 3 and with the lattice of . Because of the index 3 of in , there are three conjugate subgroups of the type with their fourfold axes directed along the z, x and y directions of the cubic space group . The group is a subgroup of index 2 of . Therefore, two domain states (with opposite polar axes) belong to each of the subgroups of the space-group type .

In reality, i.e. without the PCA, the additional degree of freedom in tetragonal space groups leads to a metrical inequivalence of the directions of the fourfold axis and the axes perpendicular to it, due to the spontaneous strain. If this `true physical situation' is considered, the 90° angles between the fourfold axes of the domains of different pairs will change slightly because of the ferroelastic tetragonal deformation with subsequent small rotations of the domains. The antiparallel nature of each of the three domain pairs, however, is preserved because they exhibit the same symmetry state, i.e. belong to the same space group.

The space groups and in the phase transition in perovskites, cf. Example 1.2.7.2.4, are translationengleiche. Therefore, , and the coset decomposition of the space groups corresponds to the coset decomposition of their point groups, cf. Section 1.2.6.2. Indeed, in the literature the perovskite transition is nearly always treated using point groups. Other than the microscopic description by space groups, the description by point groups is called the macroscopic or continuum description. Because there are no translations involved, the continuum approach does not require the PCA at all; the point group is for any translations of phase B a subgroup of the point group . However, the spontaneous strains will appear in this ferroelastic transition and will cause the complications mentioned at the end of Section 1.2.7.1 and in the paragraph preceding Lemma 1.2.7.2.3. For non-ferroelastic translationengleiche transitions, the application of point groups and of space groups yields equivalent results.

Example 1.2.7.2.4. presents an opportunity to mention another feature of this phase transition which, however, will not be discussed further. The group is not a maximal subgroup of but with of index 3 in and of index 2 in . Such `intermediate' domain states between A and B, like the domain state with the space group , are called secondary domain states for thermodynamic reasons. They do not appear in the transition but is the symmetry of the spontaneous strain in the domain state with space group . Accidentally, in the present example holds. Secondary domain states are treated by Janovec & Přívratská (2003).

In Example 1.2.7.2.4, a phase transition was discussed which involves only translationengleiche group–subgroup relations and, hence, only orientational relations between the domains. The following two examples treat klassengleiche transitions, i.e. is a klassengleiche subgroup of , and translational domain structures, also called translation twins, may appear.

Translational domain structures consist of domains which are parallel, i.e. have the same orientation of their structures (and thus of their lattices) but differ in their location because of the loss of translations of the parent phase in the phase transition. The origins of the larger unit cells of the phase B with subgroup may coincide with any of the origins of the smaller unit cells of the parent structure A with space group . Again the number of such domain states is equal to the index of in , ; the number of symmetry states is .

These shifts do not show up in the macroscopic properties of the domains. Indeed, one is normally neither interested in those translations of which are lost in the transition to the subgroup , nor in the position of the conventional origin of relative to that of but only in the orientation of the domain states . If so, the observed relations are not governed by the space groups and but by and Hermann's group , , cf. Lemma 1.2.8.1.2. The group is uniquely determined as the space group with the translations of and the point-group operations of . The group can thus be characterized as that translationengleiche subgroup of which is at the same time a klassengleiche supergroup of . This group plays a role in the practical treatment of domains. It was applied to domain structures first by Janovec (1976).

In the current literature, the following considerations are mostly restricted to the point groups of the phases involved. In the following, the use of Hermann's group is discussed in parallel with the normal use of the point groups. The (admittedly rather abstract) discussion may thus be unfamiliar to the reader. Nevertheless, it is offered here because it opens up the possibility of treating phase transitions on a microscopic or atomistic level, whereas the point-group approach can only deal with the continuum or macroscopic aspect. The microscopic approach is necessary in particular when discussing domain boundaries, which will not be done here.

In Example 1.2.7.3.1 of AuCu₃, because is a klassengleiche subgroup of . Therefore, and all four domain states belong to the same orientation state. This is obvious visually, because, as stated above, all four domain states are parallel and only shifted against each other.

Lemma 1.2.7.3.3 is the microscopic formulation of the (macroscopic) continuum treatment of phase transitions and forms the bridge from the (macroscopic) continuum to the (microscopic) atomistic approach to phase transitions.

Up to now, the examples have been concerned either with translationengleiche or with klassengleiche transitions only. In this section, the domain structure of a general transition will be considered, i.e. a transition where is a general subgroup of . General subgroups are not listed in this volume but have to be derived from the maximal subgroups of each single step of the group–subgroup chain between and . In the following Example 1.2.7.4.2, the chain has two steps. The results obtained under the PCA and without it are different and, therefore, will be discussed in some detail. Example 1.2.7.4.2 further shows how the subgroup data of this volume can be used for the analysis of continuous or quasi-continuous phase transitions.

We start with a lemma for general subgroups which contains the results of Lemmata 1.2.7.3.3 and 1.2.7.3.4.

It turns out that the transition of Gd₂(MoO₄)₃ can be considered both under the PCA (allowing exact group-theoretical arguments) and under physically realistic arguments (which require certain relaxations of the group-theoretical methods). The results are different but the realistic approach can be developed by means of an increasing deviation from the PCA, starting from idealized but unrealistic considerations.

In this volume, only subgroups of finite index i are listed. However, the index i is not restricted, i.e. there is no number I with the property for any i. Subgroups with infinite index are considered in International Tables for Crystallography, Vol. E (2002).

The decisive point is that any group–subgroup chain between space groups can be split into a translationengleiche subgroup chain between the space groups and and a klassen­glei­che subgroup chain between the space groups and .

It may happen that either or holds. In particular, one of these equations must hold if is a maximal subgroup of .

The following lemma holds for space groups but not for arbitrary groups of infinite order.

This number of subgroups can be further specified, see Chapter 1.5 . Although for each index i the number of subgroups is finite, the number of all subgroups with finite index is infinite because there is no upper limit for the number i.

Even the set of all maximal subgroups of finite index is not finite, as can be seen from the following lemma.

An index of occurs only for isomorphic subgroups of tetragonal, trigonal and hexagonal space groups when the basis vectors are enlarged to pa, pb. An index of occurs for and only for isomorphic subgroups of cubic space groups with cell enlargements of pa, pb, pc ().

This lemma means that a subgroup of, say, index 6 cannot be maximal. Moreover, because of the infinite number of primes, the set of all maximal subgroups of a given space group cannot be finite.

There are even stronger restrictions for maximal non-isomorphic subgroups.

This lemma can be specified further:

There are also lemmata for the number of subgroups of a certain index. The most important are:

Examples are:

subgroups of index 2 for , No. 13, and , No. 196;

subgroup of index 2 for , No. 14, and , No. 143; ;

subgroups of index 2 for , No. 6, and , No. 2;

subgroups of index 2 for , No. 47.

Nevertheless, their listing is possible in the form of infinite series. The series are specified by parameters.

The series of maximal isomorphic subgroups are discussed in Section 2.1.5 .