International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2011). Vol. A1, ch. 1.4, pp. 27-40   | 1 | 2 |
https://doi.org/10.1107/97809553602060000793

Chapter 1.4. The mathematical background of the subgroup tables

Gabriele Nebea*

aLehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, D-52062 Aachen, Germany
Correspondence e-mail: nebe@math.rwth-aachen.de

This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. The algebraic concepts of vector spaces, the affine space and the affine group are defined and discussed. A section on groups with special emphasis on actions of groups on sets, the Sylow theorems and the isomorphism theorems follows. After the definition of space groups, their maximal subgroups are considered and the theorem of Hermann is derived. It is shown that a maximal subgroup of a space group has a finite index and is a space group again. From the proof that three-dimensional space groups are soluble groups, it follows that the indices of their maximal subgroups are prime powers. Special considerations are devoted to the subgroups of index 2 and 3. Furthermore, a maximal subgroup is an isomorphic subgroup if its index is larger than 4. In addition, more special quantitative results on the numbers and indices of maximal subgroups of space groups are derived. The abstract definitions and theorems are illustrated by several examples and applications.

1.4.1. Introduction

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This chapter gives a brief introduction to the mathematics involved in the determination of the subgroups of space groups. To achieve this we have to detach ourselves from the geometric point of view in crystallography and introduce more abstract algebraic structures, such as coordinates, which are well known in crystallography and permit the formalization of symmetry operations, and also the abstract notion of a group, which allows us to apply general theorems to the concrete situation of (three-dimensional) space groups.

This algebraic point of view has the following advantages:

 (1) Geometric problems can be treated by algebraic calculations. These calculations can be dealt with by well established procedures. In particular, the use of computers and advanced programs enables one to solve even difficult problems in a comparatively short time. (2) The mappings form groups in the mathematical sense of the word. This means that the very powerful methods of group theory may be applied successfully. (3) The procedures for the solution may be developed to a great extent independently of the dimension of the space.

In Section 1.4.2, a basis is laid down which gives the reader an understanding of the algebraic point of view of the crystal space (or point space) and special mappings of this space onto itself. The set of these mappings is an example of a group. For a closer connection to crystallography, the reader may consult Section 8.1.1 of International Tables for Crystallography Volume A (2005) (abbreviated as IT A) or the book by Hahn & Wondratschek (1994).

Section 1.4.3 gives an introduction to abstract groups and states the important theorems of group theory that will be applied in Section 1.4.4 to the most important groups in crystallography, the space groups. In particular, Section 1.4.4 treats maximal subgroups of space groups which have a special structure by the theorem of Hermann. In Section 1.4.5, we come back to abstract group theory stating general facts about maximal subgroups of groups. These general theorems allow us to calculate the possible indices of maximal subgroups of three-dimensional space groups in Section 1.4.6. The next section, Section 1.4.7, deals with the very subtle question of when these maximal subgroups of a space group are isomorphic to this space group. In Section 1.4.8 minimal supergroups of space groups are treated briefly.

1.4.2. The affine space

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1.4.2.1. Motivation

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The aim of this section is to give a mathematical model for the point space' (also known in crystallography as direct space' or crystal space') which contains the positions of atoms in crystals (the so-called points'). This allows us in particular to describe the symmetry groups of crystals and to develop a formalism for calculating with these groups which has the advantage that it works in arbitrary dimensions. Such higher-dimensional spaces up to dimension 6 are used, for example, for the description of quasicrystals and incommensurate phases. For example, the more than 29 000 000 crystallographic groups up to dimension 6 can be parameterized, constructed and identified using the computer package [CARAT]: Crystallographic AlgoRithms And Tables, available from http://wwwb.math.rwth-aachen.de/carat/index.html (for a description, see Opgenorth et al., 1998).

As well as the points in point space, there are other objects, called vectors'. The vector that connects the point P to the point Q is usually denoted by . Vectors are usually visualized by arrows, where parallel arrows of the same length represent the same vector.

Whereas the sum of two points P and Q is not defined, one can add vectors. The sum of two vectors and is simply the sum of the two arrows. Similarly, multiplication of a vector by a real number can be defined.

All the points in point space are equally good, but among the vectors one can be distinguished, the null vector . It is characterized by the property that for all vectors .

Although the notion of a vector seems to be more complicated than that of a point, we introduce vector spaces before giving a mathematical model for the point space, the so-called affine space, which can be viewed as a certain subset of a higher-dimensional vector space, where the addition of a point and a vector makes sense.

1.4.2.2. Vector spaces

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We shall now exploit the advantage of being independent of the dimensionality. The following definitions are independent of the dimension by replacing the specific dimensions 2 for the plane and 3 for the space by an unspecified integer number . Although we cannot visualize four- or higher-dimensional objects, we can describe them in such a way that we are able to calculate with such objects and derive their properties.

Algebraically, an n-dimensional (real) vector v can be represented by a column of n real numbers. The n-dimensional real vector space is then (In crystallography n is normally 3.) The entries are called the coefficients of the vector . On one can naturally define an addition, where the coefficients of the sum of two vectors are the corresponding sums of the coefficients of the vectors. To multiply a vector by a real number, one just multiplies all its coefficients by this number. The null vector can be distinguished, since for all .

The identification of a concrete vector space with the vector space can be done by choosing a basis of . A basis of is any tuple of n vectors such that every vector of can be written uniquely as a linear combination of the basis vectors: . Whereas a vector space has many different bases, the number n of vectors of a basis is uniquely determined and is called the dimension of . The isomorphism (see Section 1.4.3.4 for a definition of isomorphism) between and maps the vector to its coefficient column with respect to the chosen basis . The mapping respects addition of vectors and multiplication of vectors with real numbers. Moreover, is a bijective mapping, which means that for any coefficient column there is a unique vector with . Therefore one can perform all calculations using the coefficient columns.

An important concept in mathematics is the automorphism group of an object. In general, if one has an object (here the vector space ) together with a structure (here the addition of vectors and the multiplication of vectors with real numbers), its automorphism group is the set of all one-to-one mappings of the object onto itself that preserve the structure.

A bijective mapping of the vector space into itself satisfying for all and for all real numbers and all vectors is called a linear mapping and the set of all these linear mappings is the linear group of . To know the image of under a linear mapping it suffices to know the images of the basis vectors under , since . Writing the coefficient columns of the images of the basis vectors as columns of a matrix [i.e. , ], then the coefficient column of with respect to the chosen basis is just . Note that the matrix of a linear mapping depends on the basis of . The matrix that corresponds to the composition of two linear mappings is the product of the two corresponding matrices. We have thus seen that the linear group of a vector space of dimension n is isomorphic to the group of all invertible matrices via the isomorphism that associates to a linear mapping its corresponding matrix (with respect to the basis ). This means that one can perform all calculations with linear mappings using matrix calculations.

In crystallography, the translation-vector space has an additional structure: one can measure lengths and angles between vectors. An n-dimensional real vector space with such an additional structure is called a Euclidean vector space, . Its automorphism group is the set of all (bijective) linear mappings of onto itself that preserve lengths and angles and is called the orthogonal group of . If one chooses the basis to be the unit vectors (which are orthogonal vectors of length 1), then the isomorphism above maps the orthogonal group onto the set of all matrices A with , the unit matrix. T denotes the transposition operator, which maps columns to rows and rows to columns.

1.4.2.3. The affine space

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In this section we build up a model for the point space'. Let us first assume . Then the affine space may be imagined as an infinite sheet of paper parallel, let us say, to the (, ) plane and cutting the axis at in crystallographic notation. The points of have coordinates which are the coefficients of the vector from the origin to the point.

This observation is generalized by the following:

Definition 1.4.2.3.1. is an n-dimensional affine space.

If then the vector is defined as the difference (computed in the vector space ). The set of all with forms an n-dimensional vector space which is called the underlying vector space . Omitting the last coefficient, we can identify with . As the coordinates already indicate, the sets as well as can be viewed as subsets of . Computed in , the sum of two elements in is again in , since the last coefficient of the sum is and the sum of a point and a vector is again a point in (since the last coordinate is ), but the sum of two points does not make sense.

1.4.2.4. The affine group

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The affine group of geometry is the set of all mappings of the point space which fulfil the conditions

 (1) parallel straight lines are mapped onto parallel straight lines; (2) collinear points are mapped onto collinear points and the ratio of distances between them remains constant.

In the mathematical model, the affine group is the automorphism group of the affine space and can be viewed as the set of all linear mappings of that preserve .

Definition 1.4.2.4.1. The affine group is the subset of the set of all linear mappings with . The elements of are called affine mappings.

Since is linear, it holds that Hence an affine mapping also maps into itself.

Since the first n basis vectors of the chosen basis lie in and the last one in , it is clear that with respect to this basis the affine mappings correspond to matrices of the formThe linear mapping induced by on which is represented by the matrix will be referred to as the linear part of . The image of a point P with coordinates can easily be found as

If one has a way to measure lengths and angles (i.e. a Euclidean metric) on the underlying vector space , one can compute the distance between P and Q as the length of the vector and the angle determined by P, Q and R with vertex Q is obtained from . In this case, is the Euclidean point space, .

An affine mapping of the Euclidean point space is called an isometry if its linear part is an orthogonal mapping of the Euclidean vector space . The set of all isometries in is called the Euclidean group and denoted by . Hence is the set of all distance-preserving mappings of onto itself. The isometries are the affine mappings with matrices of the form where the linear part W belongs to the orthogonal group of .

Special isometries are the translations, the isometries where the linear part is , with matrix The group of all translations in is the translation subgroup of and is denoted by . Note that composition of two translations means addition of the translation vectors and is iso­morphic to the translation vector space .

1.4.3. Groups

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1.4.3.1. Groups

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The affine group is only one example of the more general concept of a group. The following axiomatic definition sometimes makes it easier to examine general properties of groups.

Definition 1.4.3.1.1. A group is a set with a mapping , called the composition law or multiplication of , satisfying the following three axioms:

 (i) for all (associative law). (ii) There is an element called the unit element of with for all . (iii) For all , there is an element , called the inverse of , with .

Normally the symbol · is omitted, hence the product is just written as and the set is called a group.

One should note that in particular property (i), the associative law, of a group is something very natural if one thinks of group elements as mappings. Clearly the composition of mappings is associative. In general, one can think of groups as groups of mappings as explained in Section 1.4.3.2.

A subset of elements of a group which themselves form a group is called a subgroup:

Definition 1.4.3.1.2. A non-empty subset is called a subgroup of (abbreviated as ) if for all .

The affine group is an example of a group where is given by the composition of mappings. The unit element is the identity mapping given by the matrix which also represents the translation by the vector . The composition of two affine mappings is again an affine mapping and the inverse of an affine mapping has matrix Since the inverse of an isometry and the composition of two isometries are again isometries, the set of isometries is a subgroup of the affine group . The translation subgroup is a subgroup of .

Any vector space is a group with the usual vector addition as composition law. Therefore is also a group.

Remarks

 (i) For every group , the set consisting only of the unit element of is a subgroup of called the trivial subgroup . (ii) If is a subgroup of and is a subgroup of the group , then is a subgroup of . (iii) If and are subgroups of the group , then the intersection is also a subgroup of . (iv) If is a subset of the group , then the smallest subgroup of containing S is denoted by and is called the subgroup generated by S. The elements of S are called the generators of this group. It is convenient not to list all the elements of a group but just to give generators of (this also applies to finite groups).

Example 1.4.3.1.3

A well known group is the addition group of integers where · is normally denoted by + and the unit element is 0. The group is generated by . Other generating sets are for example or . Taking two integers which are divisible by some fixed integer , then the sum and the negatives and are again divisible by p. Hence the set of all integers divisible by p is a subgroup of . It is generated by .

Definition 1.4.3.1.4. The order of a group is the number of elements in the set .

Most of the groups in crystallography, for example , , , have infinite order.

Groups that are generated by one element are called cyclic. The cyclic group of order n is called . (We prefer to use three letters to denote the mathematical names of frequently occurring groups, since the more common symbol could possibly cause confusion with the Schoenflies symbol .)

The group is not generated by a finite set.

These two groups and have the property that for all elements and in the group it holds that . Hence these two groups are Abelian in the sense of the following:

Definition 1.4.3.1.5. The group is called Abelian if for all .

1.4.3.2. Actions of groups on sets

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The affine group is defined via its action on the affine space . In general, the greatest significance of groups is that they act on sets.

Definition 1.4.3.2.1. Let be a group. A non-empty set M is called a (left) -set if there is a mapping satisfying the following conditions:

 (i) for all and . (ii) for all .

If M is a -set, one also says that acts on M.

Example 1.4.3.2.2

 (a) The affine space is a -set for the affine group . (b) is an -set, where acts via the linear parts. (c) is also a group and acts on by translations for , . (d) If is a subgroup of the group , then is a -set where is the usual composition law. In particular, each group is a -set and hence every group can be viewed as a group of mappings from onto .

Definition 1.4.3.2.3. Let be a group and M a -set. If , then the set is called the orbit of m under .

The -set M is called transitive if for any consists of a single orbit under .

If then the stabilizer of m in is .

For a space group and a point P in the point space, the stabilizer is called the site-symmetry group of P with respect to .

The kernel of the action of on M is the intersection of the stabilizers of all elements in M, M is called a faithful -set and the action of on M is also called faithful if the kernel of the action is trivial, .

Note that any space group acts faithfully on the point space.

Remarks

 (i) If , then their orbits are either equal or disjoint. For if there is an element , then by the axioms of -sets , hence every element in the orbit of is of the form and therefore lies in the orbit of . Hence the set of orbits gives a partition of M into disjoint sets. If M is a finite set, then its order is the sum of the lengths of the different orbits. (ii) is a subgroup of , since for , the product . (iii) If , then .

Example 1.4.3.2.4 (Example 1.4.3.2.2 continued)

 (a) is a transitive -set. This is a mathematical expression of the fact that in point space no point is distinguished. (b) The -set decomposes into two orbits and . The kernel of the action of on is the translation subgroup . (c) acts transitively on . The kernel of the action only consists of the zero vector .

We now introduce some terminology for groups which is nicely formulated using -sets.

Definition 1.4.3.2.5. The orbit of under the action of the subgroup is the right coset (cf. IT A, Section 8.1.5 ). Analogously one defines a left coset as and denotes the set of left cosets by .

If the number of left cosets (which is always equal to the number of right cosets) of in is finite, then this number is called the index of in . If this number is infinite, one says that the index of in is infinite.

Example 1.4.3.2.6

is a coset of in , namely If one thinks of as an infinite sheet of paper and puts uncountably many such sheets of paper (one for each real number) one onto the other, one gets the whole -space .

Remark. The set of left cosets is a left -set with the operation for all . The kernel of the action is the intersection of all subgroups of that are conjugate to and is called the core of : .

We now assume that is finite. Let be a subgroup of . Then the set is partitioned into left cosets of , , where is the index of in . Since the orders of the left cosets of are all equal to the order of , one gets

Theorem 1.4.3.2.7. (Theorem of Lagrange.) Let be a subgroup of the finite group . Then In particular, the order of any subgroup of and also the index of any subgroup of are divisors of the group order .

The -set is only a special case of a -set. It is a transitive -set. If is a transitive -set, then the mapping , is a bijection (in fact an isomorphism of -sets in the sense of Definition 1.4.3.4.1 below). Therefore, the number of elements of M, which is the length of the orbit of m under , equals the index of the stabilizer of m in , whence one gets the following generalization of the theorem of Lagrange:

Theorem 1.4.3.2.8. Let be a finite group and M be a -set. Then for all .

The point group acts on the finite set M of ideal crystal faces. Then the length of the orbit (the number of equivalent crystal faces) times the order of the face-symmetry group is the order of the point group.

Up to now, we have only considered the action of upon via multiplication. There is another natural action of on itself via conjugation: defined by for all group elements and elements m in the -set . The stabilizer of m is called the centralizer of m in , If is a set of group elements, then the centralizer of M is the intersection of the centralizers of the elements in M:

Definition 1.4.3.2.9.  also acts on the set of all subgroups of by conjugation, . The stabilizer of an element is called the normalizer of and denoted by . is called a normal subgroup of (denoted as ) if .

Remarks

 (i) Let . Then the index of the normalizer of in is the number of subgroups of that are conjugate to . Since always normalizes itself [hence is a subgroup of ], the index of the normalizer divides the index of . (ii) If is Abelian, then the conjugation action of is trivial, hence each subgroup of is a normal subgroup. (iii) The group itself and also the trivial subgroup are always normal subgroups of .

Normal subgroups play an important role in the investigation of groups. If is a normal subgroup, then the left coset equals the right coset for all , because .

The most important property of normal subgroups is that the set of left cosets of in forms a group, called the factor group , as follows: The set of all products of elements of two left cosets of again forms a left coset of . Let . Then This defines a natural product on the set of left cosets of in which turns this set into a group. The unit element is .

Hence the philosophy of normal subgroups is that they cut the group into pieces, where the two pieces and are again groups.

Example 1.4.3.2.10. The group is Abelian. For any number , the set is a subgroup of . Hence is a normal subgroup of . The factor group inherits the multiplication from the multiplication in , since for all . If p is a prime number, then all elements in have a multiplicative inverse, and therefore is a field, the field with p elements.

Proposition 1.4.3.2.11. Let be a normal subgroup of the group and . Then the set is a subgroup of .

Proof. Let , . Thenwhere , since is a subgroup of , and , since is a normal subgroup of . QED

1.4.3.3. The Sylow theorems

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A nice application of the notion of -sets are the three theorems of Sylow. By Theorem 1.4.3.2.7, the order of any subgroup of a group divides the order of . But conversely, given a divisor d of , one cannot predict the existence of a subgroup of with . If is a prime power that divides , then the following theorem says that such a subgroup exists.

Theorem 1.4.3.3.1. (Sylow) Let be a finite group and p be a prime such that divides the order of . Then possesses m subgroups of order , where satisfies .

In particular this theorem implies that for every prime power that divides the order of the finite group , the group has a subgroup whose order is this prime power. This is not true for composite numbers. For instance, the alternating group of order 12 (Hermann–Mauguin notation 23) has no subgroup of order 6. This group has three subgroups of order 2, a unique subgroup of order 4 = 22 and four subgroups of order 3. The group (Hermann–Mauguin notation ) also has order 12 but seven subgroups of order 2, three subgroups of order 4 and a unique subgroup of order 3.

Theorem 1.4.3.3.2. (Sylow) If for some prime p not dividing s, then all subgroups of order of are conjugate in . Such a subgroup of order is called a Sylow p-subgroup.

Combining these two theorems with Theorem 1.4.3.2.8, one gets Sylow's third theorem:

Theorem 1.4.3.3.3. (Sylow) The number of Sylow p-sub­groups of is and divides the order of .

Proofs of the three theorems above can be found in Ledermann (1976), pp. 158–164, or in Ledermann & Weir (1996), pp. 155–161.

1.4.3.4. Isomorphisms

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If one wants to compare objects such as groups or -sets, to be able to say when they should be considered equal, the concept of isomorphisms should be used:

Definition 1.4.3.4.1. Let and be groups and M and N be -sets.

 (a) A homomorphism is a mapping of the set into the set respecting the composition law i.e. for all . If is bijective, it is called an isomorphism and one says is isomorphic to (). If is the unit element of , then the set of all pre-images of is called the kernel of : . An isomorphism is called an automorphism of . (b) M and N are called isomorphic -sets if there is a bijection with for all .

Example 1.4.3.4.2. In Example 1.4.3.1.3, the group homomorphism defined by is a group isomorphism (from the group onto its subgroup ).

Example 1.4.3.4.3. For any group element , conjugation by defines an automorphism of . In particular, if is a subgroup of , then and its conjugate subgroup are isomorphic.

Philosophy: If and are isomorphic groups, then all group-theoretical properties of and are the same. The calculations in can be translated by the isomorphism to calculations in . Sometimes it is easier to calculate in one group than in the other and translate the result back via the inverse of the isomorphism. For example, the isomorphism between and in Section 1.4.2 is an isomorphism of groups. It even respects scalar multiplication with real numbers, so in fact it is an isomorphism of vector spaces. While the composition of translations is more concrete and easier to imagine, the calculation of the resulting vector is much easier in . The concept of isomorphism says that you can translate to the more convenient group for your calculations and translate back afterwards without losing anything.

Note that a homomorphism is injective, hence an isomorphism onto its image, if and only if its kernel is trivial .

Example 1.4.3.4.4

The mapping μ from the space of translation vectors into the affine group defined byis a homomorphism of the group into . The kernel of this homomorphism is and the image of the mapping is the translation subgroup of . Hence the groups and are isomorphic.

The affine group acts (as group of group automorphisms) on the normal subgroup via conjugation: . We have seen already in Example 1.4.3.2.4 (b) that it also acts (as a group of linear mappings) on . The mapping is an isomorphism of -sets.

1.4.3.5. Isomorphism theorems

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[cf. Ledermann (1976), pp. 68–73, or Ledermann & Weir (1996), pp. 85–92.]

Remark. If is a homomorphism and is a normal subgroup of , then the pre-image is a normal subgroup of . In particular, it holds that .

Hence the factor group is a well defined group. The following theorem says that this group is isomorphic to the image of :

Theorem 1.4.3.5.1. (First isomorphism theorem.) Let be a homomorphism of groups. Then defined by is an isomorphism between the factor group and the image group of , which is a subgroup of .

For instance, if is a space group and is mapping any elementto its linear part W, then the kernel of is the translation group of and the image is the point group of . The theorem says that the point group is isomorphic to the factor group .

Theorem 1.4.3.5.2. (Third isomorphism theorem.) Let be a normal subgroup of the group and be an arbitrary subgroup of . Then is a normal subgroup of and (For the definition of the group see Proposition 1.4.3.2.11.)

Definition 1.4.3.5.3. A subgroup is a characteristic sub­group if for all automorphisms of .

Remarks

 (a) If is a finite Abelian group and is a Sylow p-subgroup of , then , because is the only subgroup of of order . (b) If is any group and , then is also a normal subgroup of : for define the mapping , . Then is an automorphism of and since is characteristic in . (c) If , then , since the conjugation with any element of induces an automorphism of .

1.4.3.6. An example

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Let us consider the tetrahedral group, Schoenflies symbol , which is defined as the symmetry group of a tetrahedron. It permutes the four apices of the tetrahedron and hence every element of defines a bijection of onto itself. The only element that fixes all the apices is . Therefore the set V is a faithful -set. Let us calculate the order of . Since there are elements in that map the first apex onto each one of the other apices, V is a transitive -set. Let be the stabilizer of . By Theorem 1.4.3.2.8, . The group is generated by the threefold rotation around the diagonal' of the tetrahedron through and the reflection at the symmetry plane of the tetrahedron which contains the edge . In particular, acts transitively on the set . The stabilizer of in is the cyclic group generated by . (The Schoenflies notation for is and the Hermann–Mauguin symbol is m.) Therefore and . In fact, we have seen that is isomorphic to the group of all bijections of V onto itself, which is the symmetric group of degree 4 and the group is the symmetric group on . The Schoenflies notation for is and its Hermann–Mauguin symbol is .

In general, let be a natural number. Then the group of all bijective mappings of the set onto itself is called the symmetric group of degree n and denoted by The alternating group is the normal subgroup consisting of all even permutations of .

Let us construct a normal subgroup of . The tetrahedral group contains three twofold rotations around the three axes of the tetrahedron through the midpoints of opposite edges. Since permutes these three axes and hence conjugates the three rotations into each other, the group generated by these three rotations is a normal subgroup of . Since these three rotations commute with each other, the group is Abelian. Now and hence (in Schoenflies notation) (Hermann–Mauguin symbol) is of order 4. There are three normal subgroups of order 2 in , namely for . The factor group is again of order 2. Since all groups of order 2 are cyclic, . The set is the set of all products of elements from the two normal subgroups and , hence is isomorphic to the direct product in the sense of the following definition.

Definition 1.4.3.6.1. [cf. Ledermann (1976), Section 13, or Ledermann & Weir (1996), Section 2.7.] Let and be two groups. Then the direct product is the group with multiplication .

Let us return to the example above. The centralizer of one of the three rotations, say of , is of index 3 in and hence a Sylow 2-subgroup of with order 8. Following Schoenflies, we will denote this group by (another Schoenflies symbol for this group is and its Hermann–Mauguin symbol is ).

The group above is contained in . It is its own centralizer in : . Therefore, the factor group acts faithfully (and transitively) on the set . The stabilizer of is the subgroup constructed above. Using this, one easily sees that .

Another normal subgroup in is the set of all rotations in . This group contains the normal subgroup above of index 3 and is of index 2 in (and hence has order 12). It is isomorphic to , the alternating group of degree 4, and has Schoenflies symbol T and Hermann–Mauguin symbol .

1.4.4. Space groups

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1.4.4.1. Definition of space groups

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In IT A (2005) Section 8.1.6 space groups are introduced as symmetry groups of crystal patterns.

Definition 1.4.4.1.1

 (a) Let be the n-dimensional real vector space. A subset is called an (n-dimensional) lattice if there is a basis of such that (b) A crystal structure is a mapping of the Euclidean point space into the real numbers such that is an n-dimensional lattice in . (c) The Euclidean group acts on the set of mappings via for all and for all and . A space group is the stabilizer of a crystal structure ; . (d) Let be a space group. The translation subgroup of is defined as

The definition introduced space groups in the way they occur in crystallography: The group of symmetries of an ideal crystal stabilizes the crystal structure. This definition is not very helpful in analysing the structure of space groups. If is a space group, then the translation subgroup is a normal subgroup of . It is even a characteristic subgroup of , hence fixed under every automorphism of . By Definition 1.4.4.1.1, its image under the inverse of the mapping in Example 1.4.3.4.4 defined by in is an n-dimensional lattice . Since is an isomorphism from onto , the translation subgroup of is isomorphic to the lattice . In particular, one has and the subgroup , formed by the pth powers of elements in , is mapped onto . Lattices are well understood. Although they are infinite, they have a simple structure, so they can be examined algorithmically. Since they lie in a vector space, one can apply linear algebra to them.

Now we want to look at how this lattice fits into the space group . The affine group acts on by conjugation as well as on via its linear part. Similarly the space group acts on by conjugation: For and , one gets , where is the linear part of . Therefore, the kernel of this action is on the one hand the centralizer of in , on the other hand, since contains a basis of , it is equal to the kernel of the mapping , which is , hence Hence only the linear part of acts faithfully on by conjugation and linearly on . This factor group is a finite group. Let us summarize this:

Theorem 1.4.4.1.2. Let be a space group. The translation subgroup is an Abelian normal subgroup of which is its own centralizer, . The finite group acts faithfully on by conjugation. This action is similar to the action of the linear part on the lattice .

1.4.4.2. Maximal subgroups of space groups

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Definition 1.4.4.2.1. A subgroup of a group is called maximal if and for all subgroups with it holds that either or .

The translation subgroup of the space group plays a very important role if one wants to analyse the space group . Let be a subgroup of . Then has either fewer translations () or the order of the linear part of , the index of in , gets smaller (), or both happen.

Definition 1.4.4.2.2. Let be a subgroup of the space group and .

 (t) is called a translationengleiche or a t-subgroup if . (k) is called a klassengleiche or a k-subgroup if .

Remark. The third isomorphism theorem, Theorem 1.4.3.5.2, implies that if is a k-subgroup, then . Hence is a k-subgroup if and only if .

Let be a maximal subgroup of . Then we have the following preliminary situation:

Since and , one has by Proposition 1.4.3.2.11 that . Hence the maximality of implies that or . If then , hence is a t-subgroup. If , then by the third isomorphism theorem, Theorem 1.4.3.5.2, , hence is a k-subgroup. This is given by the following theorem:

Theorem 1.4.4.2.3. (Hermann) Let be a maximal subgroup of the space group . Then is either a k-subgroup or a t-subgroup.

The above picture looks as follows in the two cases:

Let be a t-subgroup of . Then and is a subgroup of . On the other hand, any subgroup of defines a unique t-subgroup of with and , namely . Hence the t-subgroups of are in bijection to the subgroups of , which is a finite group according to the remarks below Definition 1.4.4.1.1. For future reference, we note this in the following corollary:

Corollary 1.4.4.2.4. The t-subgroups of the space group are in bijection with the subgroups of the finite group .

In the case , which is the most important case in crystallography, the finite groups are isomorphic to subgroups of either (Hermann–Mauguin symbol ) or (). Here denotes the direct product (cf. Definition 1.4.3.6.1), the cyclic group of order 2, and and the symmetric groups of degree 3 or 4, respectively (cf. Section 1.4.3.6). Hence the maximal subgroups of that are t-subgroups can be read off from the subgroups of the two groups above.

An algorithm for calculating the maximal t-subgroups of which applies to all three-dimensional space groups is explained in Section 1.4.5.

The more difficult task is the determination of the maximal k-subgroups.

Lemma 1.4.4.2.5. Let be a maximal k-subgroup of the space group . Then is a normal subgroup of . Hence is an -invariant lattice.

Proof. , so every element in can be written as where and . Therefore one obtains for since is Abelian. Since and is normal in , one has . But is a product of elements in and therefore lies in the subgroup , hence . QED

The candidates for translation subgroups of maximal k-subgroups of can be found by linear-algebra algorithms using the philosophy explained at the beginning of this section: acts on by conjugation and this action is isomorphic to the action of the linear part of on the lattice via the isomorphism . Normal subgroups of contained in are mapped onto -invariant sublattices of . An example for such a normal subgroup is the group formed by the pth powers of elements of for any natural number, in particular for prime numbers . One has .

If is a maximal k-subgroup of , then is a normal subgroup of that is maximal in , which means that is a maximal -invariant sublattice of . Hence it contains for some prime number p. One may view as a finite -module and find all candidates for such normal subgroups as full pre-images of maximal -submodules of . This gives an algorithm for calculating these normal subgroups, which is implemented in the package [CARAT].

The group is an Abelian group, with the additional property that for all one has . Such a group is called an elementary Abelian p-group.

From the reasoning above we find the following lemma.

Lemma 1.4.4.2.6. Let be a maximal k-subgroup of the space group . Then is an elementary Abelian p-group for some prime p. The order of is with .

Corollary 1.4.4.2.7. Maximal subgroups of space groups are again space groups and of finite index in the supergroup.

Hence the first step is the determination of subgroups of that are maximal in and normal in , and is solved by linear-algebra algorithms. These subgroups are the candidates for the translation subgroups for maximal k-subgroups . But even if one knows the isomorphism type of , the group does not in general determine . Given such a normal subgroup that is contained in , one now has to find all maximal k-subgroups with and . It might happen that there is no such group . This case does not occur if is a symmorphic space group in the sense of the following definition:

Definition 1.4.4.2.8. A space group is called symmorphic if there is a subgroup such that and . The subgroup is called a complement of the translation subgroup .

Note that the group in the definition is isomorphic to and hence a finite group.

If is symmorphic and is a complement of , then one may take .

This shows the following:

Lemma 1.4.4.2.9. Let be a symmorphic space group with translation subgroup and an -invari­ant subgroup of (i.e. ). Then there is at least one k-subgroup with translation subgroup .

In any case, the maximal k-subgroups, , of satisfy

and

is a maximal -invariant subgroup of .

To find these maximal subgroups, , one first chooses such a subgroup . It then suffices to compute in the finite group . If there is a complement of in , then every element may be written uniquely as with , . In particular, any other complement of in is of the form . One computes . Since is a subgroup of , it holds that . Moreover, every mapping with this property defines some maximal subgroup as above. Since and are finite, it is a finite problem to find all such mappings.

If there is no such complement , this means that there is no (maximal) k-subgroup of with .

1.4.5. Maximal subgroups

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1.4.5.1. Maximal subgroups and primitive -sets

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To determine the maximal t-subgroups of a space group , essentially one has to calculate the maximal subgroups of the finite group . There are fast algorithms to calculate these maximal subgroups if this finite group is soluble (see Definition 1.4.5.2.1), which is the case for three-dimensional space groups. To explain this method and obtain theoretical consequences for the index of maximal subgroups in soluble space groups, we consider abstract groups again in this section.

For an arbitrary group , one has a fast method of checking whether a given subgroup of finite index is maximal by inspection of the -set of left cosets of in . Assume that and let with , and with , . Then the set may be written as Then permutes the lines of the rectangle above: For all and all , the left coset is equal to some for an . Hence the jth line is mapped onto the set

Definition 1.4.5.1.1. Let be a group and X a -set.

 (i) A congruence on X is a partition of X into non-empty subsets such that for all , , implies . (ii) The congruences and are called the trivial congruences. (iii) X is called a primitive -set if is transitive on X, and X has only the trivial congruences.

Hence the considerations above have proven the following lemma.

Lemma 1.4.5.1.2. Let be a subgroup of the group . Then is a maximal subgroup if and only if the -set is primitive.

The advantage of this point of view is that the groups having a faithful, primitive, finite -set have a special structure. It will turn out that this structure is very similar to the structure of space groups.

If X is a -set and is a normal subgroup of , then acts on the set of -orbits on X, hence is a congruence on X. If X is a primitive -set, then this congruence is trivial, hence or for all . This means that either acts trivially or transitively on X.

One obtains the following:

Theorem 1.4.5.1.3. [Theorem of Galois (ca 1830).] Let be a finite group and let X be a faithful, primitive -set. Assume that is an Abelian normal subgroup. Then

 (a) is a minimal normal subgroup of (i.e. for all , or ). (b) is an elementary Abelian p-group for some prime p and is a prime power. (c) and is the unique minimal normal subgroup of .

Proof. Let be an Abelian normal subgroup. Then acts faithfully and transitively on X. To establish a bijection between the sets and X, choose and define . Since is transitive, is surjective. To show the injectivity of , let with . Then , hence . But then acts trivially on X, because if then the transitivity of implies that there is an with . Then , since is Abelian. Since X is a faithful -set, this implies and therefore . This proves . Since this equality holds for all nontrivial Abelian normal subgroups of , statement (a) follows. If p is some prime dividing , then the Sylow p-subgroup of is normal in , since is Abelian. Therefore, it is also a characteristic subgroup of and hence a normal subgroup in (see the remarks below Definition 1.4.3.5.3). Since is a minimal normal subgroup of , this implies that is equal to its Sylow p-subgroup. Therefore, the order of is a prime power for some prime p and . Similarly, the set is a normal subgroup of properly contained in . Therefore, and is elementary Abelian. This establishes (b).

To see that (c) holds, let . Choose . Then . Since acts transitively, there is an such that . Hence . As above, let be any element of X. Then there is an element with . Hence . Since z was arbitrary and X is faithful, this implies that . Therefore, . Since is Abelian, one has , hence . To see that is unique, let be another normal subgroup of . Since is a minimal normal subgroup, one has , and, therefore, for , : . Hence centralizes , , which is a contradiction. QED

Hence the groups that satisfy the hypotheses of the theorem of Galois are certain subgroups of an affine group over a finite field . This affine group is defined in a way similar to the affine group over the real numbers where one has to replace the real numbers by this finite field. Then is the translation subgroup of isomorphic to the n-dimensional vector space over . The set X is the corresponding affine space . The factor group is isomorphic to a subgroup of the linear group of that does not leave invariant any nontrivial subspace of .

1.4.5.2. Soluble groups

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Definition 1.4.5.2.1. Let be a group. The derived series of is the series defined via , . The group is called the derived subgroup of . The group is called soluble if for some .

Remarks

 (i) The are characteristic subgroups of . (ii) is Abelian if and only if . (iii) is characterized as the smallest normal subgroup of , such that is Abelian, in the sense that every normal subgroup of with an Abelian factor group contains . (iv) Subgroups and factor groups of soluble groups are soluble. (v) If is a normal subgroup, then is soluble if and only if and are both soluble.

Example 1.4.5.2.2

The derived series of is(or in Hermann–Mauguin notation ) and that of is (Hermann–Mauguin notation: ).

Hence these two groups are soluble. (For an explanation of the groups that occur here and later, see Section 1.4.3.6.)

Now let be a three-dimensional space group. Then is an Abelian normal subgroup, hence is soluble. The factor group is isomorphic to a subgroup of either or and therefore also soluble. Using the remark above, one deduces that all three-dimensional space groups are soluble.

Lemma 1.4.5.2.3. Let be a three-dimensional space group. Then is soluble.

1.4.5.3. Maximal subgroups of soluble groups

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Now let be a soluble group and a maximal subgroup of finite index in . Then the set of left cosets is a primitive finite -set. Let be the kernel of the action of on X. Then the factor group acts faithfully on X. In particular, is a finite group and X is a primitive, faithful -set. Since is soluble, the factor group is also a soluble group. Let be the derived series of with . Then is an Abelian normal subgroup of . The theorem of Galois (Theorem 1.4.5.1.3) states that is an elementary Abelian p-group for some prime p and for some . Since , the order of X is the index of in . Therefore one gets the following theorem:

Theorem 1.4.5.3.1. If is a maximal subgroup of finite index in the soluble group , then its index is a prime power.

In the proof of Theorem 1.4.5.1.3, we have established a bijection between and the -set X, which is now . Taking the full pre-image of in , then one has and . Hence we have seen the first part of the following theorem:

Theorem 1.4.5.3.2. Let be a maximal subgroup of the soluble group . Then the factor group acts primitively and faithfully on , and there is a normal subgroup with and . Moreover, if is another subgroup of , with and , then is conjugate to .

Example 1.4.5.3.3

is the tetrahedral group from Section 1.4.3.2 and is the stabilizer of one of the four apices in the tetrahedron. Then and is a faithful -set which can be identified with the set of apices of the tetrahedron. The normal subgroup is the normal subgroup of Section 1.4.3.2.

Now let be as above, and take a Sylow 2-subgroup of . Then is the normal subgroup from Section 1.4.3.2 and .

These observations result in an algorithm for computing maximal subgroups of soluble groups :

 (1) compute normal subgroups [candidates for ]; (2) compute a minimal normal subgroup of ; (3) find as a complement of in .

1.4.6. Quantitative results

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This section gives estimates for the number of maximal sub­groups of a given index in space groups.

1.4.6.1. General results

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The first very easy but useful remark applies to general groups :

Remark. Let be a maximal subgroup of of finite index . Then . Hence the maximality of implies that either and is a normal subgroup of or and has i maximal subgroups that are conjugate to .

The smallest possible index of a proper subgroup is 2. It is well known and easy to see that subgroups of index 2 are normal subgroups:

Proposition 1.4.6.1.1. Let be a group and a subgroup of index . Then is a normal subgroup of .

Proof. Choose an element , . Then . Hence and therefore . Since this is also true if , the proposition follows. QED

Let be a subgroup of a group of index 2. Then is a normal subgroup and the factor group is a group of order 2. Since groups of order 2 are Abelian, it follows that the derived subgroup of (cf. Definition 1.4.5.2.1) (which is the smallest normal subgroup of such that the factor group is Abelian) is contained in . Hence all maximal subgroups of index 2 in contain . If one defines , then is an elementary Abelian 2-group and hence a vector space over the field with two elements. The maximal subgroups of are the maximal subspaces of this vector space, hence their number is , where .

This shows the following:

Corollary 1.4.6.1.2. The number of subgroups of of index 2 is of the form for some .

Dealing with subgroups of index 3, one has the following:

Proposition 1.4.6.1.3. Let be a subgroup of the group with . Then is either a normal subgroup of or and there are three subgroups of conjugate to .

Proof. is isomorphic to a subgroup of that acts primitively on . Hence either and is a normal subgroup of or , and there are three subgroups of conjugate to . QED

1.4.6.2. Three-dimensional space groups

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We now come to space groups. By Lemma l.4.5.2.3, all three-dimensional space groups are soluble. Theorem 1.4.5.3.1 says that the index of a maximal subgroup of a soluble group is a prime power (or infinite). Since the index of a maximal subgroup of a space group is always finite (see Corollary 1.4.4.2.7), we get:

Corollary 1.4.6.2.1. Let be a three-dimensional space group and a maximal subgroup. Then is a prime power.

Let be a three-dimensional space group and its point group. It is well known that the order of is of the form with or and . By Corollary 1.4.4.2.4, the t-subgroups of are in one-to-one correspondence with the subgroups of . Let us look at the t-subgroups of of index 3. It is clear that has no subgroup of index 3 if , since the index of a subgroup divides the order of the finite group by the theorem of Lagrange. If , then any subgroup of of index 3 has order and hence is a Sylow 2-subgroup of . Therefore there is such a subgroup of index 3 in by the first theorem of Sylow, Theorem 1.4.3.3.1. By the second theorem of Sylow, Theorem 1.4.3.3.2, all these Sylow 2-subgroups of are conjugate in . Therefore, by Proposition 1.4.6.1.3, the number of these groups is either 1 or 3:

Corollary 1.4.6.2.2. Let be a three-dimensional space group.

If the order of the point group of is not divisible by 3 then has no t-subgroups of index 3.

If 3 is a factor of the order of the point group of , then has either one t-subgroup of index 3 (which is then normal in ) or three conjugate t-subgroups of index 3.

1.4.7. Qualitative results

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1.4.7.1. General theory

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In this section, we want to comment on the very subtle question of deciding whether two space groups and are isomorphic.

This problem can be treated in several stages:

Let and be space groups. Since the translation subgroups are characteristic subgroups of (the maximal Abelian normal subgroup of finite index), each isomorphism induces isomorphisms of the corresponding translation subgroups (by restriction) as well as of the point groups It is convenient to view as a lattice on which the point group acts as group of linear mappings (cf. the start of Section 1.4.4). Then the isomorphism is an isomorphism of -sets, where acts on via conjugation and on via Since and centralizes itself, this action is well defined, i.e. independent of the choice of the coset representative .

The following theorem will show that the isomorphism of sufficiently large factor groups of and implies a near' isomorphism of the space groups themselves. To give a precise formulation we need one further definition.

Definition 1.4.7.1.1. For define which is the set of all rational numbers for which the denominator is prime to d. For the space group let be the group , where i.e. one allows denominators that are prime to d in the translation subgroup.

One has the following:

Theorem 1.4.7.1.2. Let and be two space groups with point groups of order . Let denote the set of normal subgroups of having finite index in . Then the following three conditions are equivalent:

 (i) There are normal subgroups with and with if and if (). (ii) . (iii) There is a bijection such that for all .

For a proof of this theorem, see Finken et al. (1980).

Remark. If are three- or four-dimensional space groups, the isomorphism in (ii) already implies the isomorphism of and , but there are counterexamples for dimension 5.

1.4.7.2. Three-dimensional space groups

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Corollary 1.4.7.2.1. Let be a three-dimensional space group with translation subgroup and p be a prime not dividing the order of the point group . Let be a subgroup of of index for some . Then

 (a) is a k-subgroup. (b) is isomorphic to .

Proof:

 (a) implies that divides . Since , one obtains as a factor of . But p is not a factor of , hence and . According to the remark following Definition 1.4.4.2.2, is a k-subgroup. (b) Let . Let if and otherwise, and let . Since , one has and . By the third iso­morphism theorem, Theorem 1.4.3.5.2, it follows that By Theorem 1.4.7.1.2 (i) (ii), one has . By the remark above, this already implies that and are isomorphic. QED

Theorem 1.4.7.2.2. Let be a three-dimensional space group and be a maximal subgroup of of index . Then

 (a) is a k-subgroup. (b) is isomorphic to .

Proof. Since is soluble, the index is a prime power (see Theorem 1.4.5.3.1). If p is not a factor of , the statement follows from Corollary 1.4.7.2.1. Hence we only have to consider the cases , and , . Since 9 is not a factor of the order of any crystallographic point group in dimension 3, assertion (a) follows if the index of is divisible by 9. If is a maximal t-subgroup, then is a primitive -set for the point group of . Since the point groups of dimension 3 have no primitive -sets of order divisible by 8, assertion (a) also follows if the index of is divisible by 8.

For all three-dimensional space groups , the module [where is identified with the corresponding lattice in as in Section 1.4.4] is not simple as a module for the point group . [It suffices to check this property for the two maximal point groups () and .] This means that is not a maximal -invariant sublattice of . Since the translation subgroup of a maximal k-subgroup of index equal to a power of 2 in is a maximal -invariant subgroup of that contains , one now finds that has no maximal k-subgroup of index 8.

Now assume that . By Corollary 1.4.7.2.1, one only needs to deal with groups such that the order of the point group is divisible by 3. is isomorphic to a subgroup of or . If is a subgroup of , then is simple and is of index 27 in [with ]. It turns out that is isomorphic to in these cases. If does not contain a subgroup isomorphic to , then the maximality of implies that is of index 3 in . Hence in this case. QED

Corollary 1.4.7.2.3. Let be a maximal subgroup of the three-dimensional space group .

 (a) If the index of is a power of 2, then or 4. (b) If 3 is a factor of the order of the point group and the index of is a power of 3, then or 27. For , is necessarily isomorphic to (by Theorem 1.4.7.2.2).

This interesting fact explains why there are no maximal subgroups of index 8 in a three-dimensional space group. If there is a maximal subgroup of a three-dimensional space group of index 9, then the order of the point group of is not divisible by three and the subgroup is a k-subgroup and isomorphic to .

In particular, there are no maximal subgroups of index 9 for trigonal, hexagonal or cubic space groups, whereas there are such subgroups of tetragonal space groups.

1.4.8. Minimal supergroups

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For several problems, for example for the prediction of a phase transition or in the search for overlooked symmetry in crystal-structure determinations etc., it is helpful to know all space groups containing a given space group , which means that . Then is called a supergroup of . Note that – in contrast to subgroups – the supergroups containing a space group of finite index need not be space groups. For instance, the one-dimensional translation grouphas a supergroup of index 2 isomorphic to which is not a subgroup of the one-dimensional affine group. is Abelian but has an element of finite order, so cannot be a space group. For the applications in crystallography, we are restricted to those supergroups of that are again space groups.

Definition 1.4.8.1. Let be a space group that is a supergroup of the space group and .

 (i) is called a translationengleiche or a t-supergroup if . (ii) is called a klassengleiche or a k-supergroup if .

Clearly is a t-supergroup (or k-supergroup, respectively) of if and only if is a t-subgroup (or k-subgroup) of and the theorem of Hermann implies the following:

Theorem 1.4.8.2. (Theorem of Hermann.) Let and be space groups such that is a minimal supergroup of . Then is either a k-supergroup or a t-supergroup.

The determination of the k-supergroups of a given space group is the easier task. For instance, if is a symmorphic space group then all its k-supergroups are also symmorphic. This is not true for k-subgroups of .

Theorem 1.4.8.3. Let be a k-supergroup of the space group . Then . If is a minimal k-supergroup of then the translation lattice of is an -invariant lattice that contains as a maximal sublattice.

Proof. Let be the translation subgroup of . Then and is isomorphic to the point group , which is a finite group. By the isomorphism theorem Therefore, group generated by and is a subgroup of containing with the same index and, therefore, . Moreover, contains and hence contains . Since is a normal subgroup of , the space group acts on by conjugation and therefore is invariant. If there is an invariant lattice such that , then, applying the isomorphism μ from Example 1.4.3.4.4, the group is a space group with . Hence the minimality of the supergroup implies that is an -invariant lattice that contains as a maximal sublattice. QED

As for maximal k-subgroups, the index of a minimal k-supergroup of is a prime power and for each prime p there is some such that has a minimal k-supergroup of index . Because of the infinite number of prime numbers, a space group has infinitely many minimal k-supergroups, but there are only finitely many minimal k-supergroups containing a given space group of given index.

This is different for t-supergroups, as the following example shows.

Example 1.4.8.4

Letbe the two-dimensional translation group . Then for all the groupis a minimal t-supergroup of containing of index 2. These groups are conjugate under the normalizer of in the affine group [see Example 3.47 in Heidbüchel (2003)], Visually this means that the discrete sets of symmetry lines of the different plane groups may be shifted by any real distance against each other or relative to an arbitrarily chosen origin. This yields uncountably many different t-supergroups of which are all of the same type.

The use of the affine and Euclidean normalizers of a space group is described in Part 15 of IT A. The affine normalizer of an n-dimensional space group acts on the set of all minimal t-supergroups of by conjugation.

Theorem 1.4.8.5.  has finitely many orbits on the set of (minimal) t-supergroups of .

Proof. Let be representatives of the -orbits on the set of n-dimensional space groups, i.e. of the types of n-dimensional space groups. For letdenote the set of all (maximal) t-subgroups of that are isomorphic to .

If is a (minimal) t-supergroup of the space group , then there is some and such that and , hence the pair of space groups for some , and .

If is a second supergroup of and such that for the same , then normalizes . Hence there are at most orbits of on the set of (minimal) t-supergroups of . QED

This proof also provides an algorithm to determine representatives of the -orbits of minimal t-supergroups of a given space group , provided that one knows representatives of all affine classes of space groups and their maximal t-subgroups. For dimensions 2 and 3 these are given in this volume. Since maximal t-subgroups of three-dimensional space groups have index 2, 3 or 4, this also holds for the minimal t-supergroups of these groups.

Up to dimension , the minimal t-supergroups and the minimal k-supergroups of a given space group can be obtained with the commands TSupergroups and KSupergroups in CARAT [see also Heidbüchel (2003)].

References

Finken, H., Neubüser, J. & Plesken, W. (1980). Space groups and groups of prime power order II. Arch. Math. 35, 203–209.
Hahn, Th. & Wondratschek, H. (1994). Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A. Sofia: Heron Press.
Heidbüchel, O. (2003). Beiträge zur Theorie der kristallographischen Raumgruppen. Aachener Beiträge zur Mathematik, 29. ISBN 3–86073-649–3.
International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer.
Ledermann, W. (1976). Introduction to Group Theory. London: Longman. (German: Einführung in die Gruppentheorie, Braunschweig: Vieweg, 1977.)
Ledermann, W. & Weir, A. J. (1996). Introduction to Group Theory. 2nd ed. Harlow: Addison Wesley Longman.
Opgenorth, J., Plesken, W. & Schulz, T. (1998). Crystallographic algorithms and tables. Acta Cryst. A54, 517–531.