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. 1.2, pp. 10-12   | 1 | 2 |

Section 1.2.3. Groups

Hans Wondratscheka*

aInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

1.2.3. Groups

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Group theory is the proper tool for studying symmetry in science. A group is a set of elements with a law of composition, by which to any two elements a third element is uniquely assigned. The symmetry group of an object is the set of all isometries (rigid motions) which map that object onto itself. If the object is a crystal, the isometries which map it onto itself (and also leave it invariant as a whole) are the crystallographic symmetry operations.

There is a huge amount of literature on group theory and its applications. The book Introduction to Group Theory (Ledermann, 1976[link]; Ledermann & Weir, 1996[link]) is recommended. The book Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A by Hahn & Wondratschek (1994[link]) describes a way in which the data of IT A can be interpreted by means of matrix algebra and elementary group theory. It may also help the reader of this volume.

1.2.3.1. Some properties of symmetry groups

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The geometric symmetry of any object is described by a group [{\cal G}]. The symmetry operations [\ispecialfonts{\sfi g}_j\in{\cal G}] are the group elements, and the set [\ispecialfonts\{{\sfi g}_j\in{\cal G}\}] 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[link]).] Groups will be designated by upper-case calligraphic script letters [{\cal G}], [{\cal H}] etc. Group elements are represented by lower-case italic sans serif letters [\ispecialfonts{\sfi g}, {\sfi h}] etc.

The result [\ispecialfonts{\sfi g}_r] of the composition of two elements [\ispecialfonts{\sfi g}_j,\,{\sfi g}_k\in{\cal G}] will be called the product of [\ispecialfonts{\sfi g}_j] and [\ispecialfonts{\sfi g}_k] and will be written [\ispecialfonts{\sfi g}_r={\sfi g}_k\,{\sfi g}_j]. 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:

  • (1) The closure, i.e. the property that the composition of any two symmetry operations results in a symmetry operation again, is always fulfilled for geometric symmetries: if [\ispecialfonts{\sfi g}_j\in{\cal G}] and [\ispecialfonts{\sfi g}_k\in{\cal G}], then [\ispecialfonts{\sfi g}_j\,{\sfi g}_k = {\sfi g}_r\in{\cal G}] also holds.

  • (2) The associative law is always fulfilled for the composition of geometric mappings. If [\ispecialfonts{\sfi g}_j,\ {\sfi g}_k,\ {\sfi g}_m \in{\cal G}], then [\ispecialfonts({\sfi g}_j\,{\sfi g}_k)\,{\sfi g}_m=] [\ispecialfonts{\sfi g}_j\, ({\sfi g}_k\,{\sfi g}_m)=] [\ispecialfonts{\sfi g}_q] for any triplet [j,\,k,\,m]. Therefore, the paren­theses are not necessary, one can write [\ispecialfonts{\sfi g}_j\,{\sfi g}_k\,{\sfi g}_m={\sfi g}_q]. In general, however, the sequence of the symmetry operations must not be changed. Thus, in general [\ispecialfonts{\sfi g}_j\,{\sfi g}_k\,{\sfi g}_m\not= {\sfi g}_j\,{\sfi g}_m\,{\sfi g}_k].

  • (3) The unit element or neutral element [\ispecialfonts{\sfi e}\in{\cal G}] is the identity operation which maps each point onto itself, i.e. leaves each point invariant.

  • (4) The isometry which reverses a given symmetry operation [\ispecialfonts{\sfi g}\in{\cal G}] is also a symmetry operation of [{\cal G}] and is called the inverse symmetry operation [\ispecialfonts{\sfi g}^{-1}] of [\ispecialfonts{\sfi g}]. It has the property [\ispecialfonts{\sfi g}\,{\sfi g}^{-1}=] [\ispecialfonts{\sfi g}^{-1}\,{\sfi g}={\sfi e}].

The number of elements of a group [{\cal G}] is called its order [|{\cal G}|]. The order of a group may be finite, for example 24 for the symmetry operations of a regular tetrahedron, or infinite, for example for any space group because of its infinite set of translations. If the relation [\ispecialfonts{\sfi g}_k\,{\sfi g}_j={\sfi g}_j\,{\sfi g}_k] is fulfilled for all pairs of elements of a group [{\cal G}], then [{\cal G}] 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.

Definition 1.2.3.1.1. A set [\ispecialfonts{\cal S}=\{{\sfi g}_p,\ {\sfi g}_q,\ \ldots\} \in{\cal G}] such that every element of [{\cal G}] can be obtained by composition of the elements of [{\cal S}] and their inverses is called a set of generators of [{\cal G}]. The elements [\ispecialfonts{\sfi g}_i\in{\cal S}] are called generators of [{\cal G}].

A group is cyclic if it consists of the unit element [\ispecialfonts{\sfi e}] and all powers of one element [\ispecialfonts{\sfi g}]: [\ispecialfonts{\cal C}({\sfi g})=\{\ \ldots\ {\sfi g}^{-3},\ {\sfi g}^{-2},\ {\sfi g}^{-1},\ {\sfi e},\ {\sfi g}^1,\ {\sfi g}^2,\ {\sfi g}^3,\ \ldots\ \}.]

If there is an integer number [n>0] with [\ispecialfonts{\sfi g}^n={\sfi e}] and n is the smallest number with this property, then the group [\ispecialfonts{\cal C}({\sfi g})] has the finite order n. Let [\ispecialfonts{\sfi g}^{-k}] with [0 \,\lt\, k \,\lt\, n] be the inverse element of [\ispecialfonts{\sfi g}^k] where n is the order of [\ispecialfonts{\sfi g}]. Because [\ispecialfonts{\sfi g}^{-k}={\sfi g}^n\,{\sfi g}^{-k}={\sfi g}^{n-k}={\sfi g}^m] with [n=m+k], the elements of a cyclic group of finite order can all be written as positive powers of the generator [\ispecialfonts{\sfi g}]. Otherwise, if such an integer n does not exist, the group [\ispecialfonts{\cal C}({\sfi g})] is of infinite order and the positive powers [\ispecialfonts{\sfi g}^{k}] are different from the negative ones [\ispecialfonts{\sfi g}^{-m}].

In the same way, from any element [\ispecialfonts{\sfi g}_j\in{\cal G}] its cyclic group [\ispecialfonts{\cal C}({\sfi g}_j)] can be generated even if [{\cal G}] is not cyclic itself. The order of this group [\ispecialfonts{\cal C}({\sfi g}_j)] is called the order of the element [\ispecialfonts{\sfi g}_j].

1.2.3.2. Group isomorphism and homomorphism

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A finite group [{\cal G}] 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[link].

Table 1.2.3.1| top | pdf |
Multiplication table of a group

The group elements [\ispecialfonts{\sfi g}\in{\cal G}] are listed at the top of the table and in the same sequence on the left-hand side; the unit element `[\ispecialfonts{\sfi e}]' is listed first. The table is thus a square array. The product [\ispecialfonts{\sfi g}_k\,{\sfi g}_j] 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 [\ispecialfonts{\sfi g}\in{\cal G}], the unit element [\ispecialfonts{\sfi e}] appears in the column of [\ispecialfonts{\sfi g}^{-1}]. If [\ispecialfonts({\sfi g})^2={\sfi e}], i.e. [\ispecialfonts{\sfi g}={\sfi g}^{-1}], [\ispecialfonts{\sfi e}] appears on the main diagonal. The multiplication table of an Abelian group is symmetric about the main diagonal.

[{\cal G}][\ispecialfonts\sfi e][\ispecialfonts\sfi a][\ispecialfonts\sfi b][\ispecialfonts\sfi c][\ldots]
[\ispecialfonts\sfi e] [\ispecialfonts\sfi e] [\ispecialfonts\sfi a] [\ispecialfonts\sfi b] [\ispecialfonts\sfi c] [\ldots]
[\ispecialfonts\sfi a] [\ispecialfonts\sfi a] [\ispecialfonts{\sfi a}^2] [\ispecialfonts{\sfi a}{\sfi b}] [\ispecialfonts{\sfi a}{\sfi c}] [\ldots]
[\ispecialfonts\sfi b] [\ispecialfonts\sfi b] [\ispecialfonts{\sfi b}{\sfi a}] [\ispecialfonts{\sfi b}^2] [\ispecialfonts{\sfi b}{\sfi c}] [\ldots]
[\ispecialfonts\sfi c] [\ispecialfonts\sfi c] [\ispecialfonts{\sfi c}{\sfi a}] [\ispecialfonts{\sfi c}{\sfi b}] [\ispecialfonts{\sfi c}^2] [\ldots]
[\vdots] [\vdots] [\vdots] [\vdots] [\vdots] [\ddots]

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.

Definition 1.2.3.2.1. Two groups are isomorphic if one can arrange the rows and columns of their multiplication tables such that these tables are equal, apart from the names or symbols of the group elements.

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 [{\cal G}] and [{\cal G}\,'] means that there is a reversible mapping [\ispecialfonts{\sfi g}_q\longleftrightarrow {\sfi g}_q'] of the elements [\ispecialfonts{\sfi g}_q\in{\cal G}] and [\ispecialfonts{\sfi g}_q'\in{\cal G}\,'] such that [\ispecialfonts({\sfi g}_j\,{\sfi g}_k)'={\sfi g}_j'\,{\sfi g}_k'] holds for any pair of indices j and k. In words:

Definition 1.2.3.2.2. Two groups [{\cal G}] and [{\cal G}\,'] are isomorphic if there is a reversible mapping of [{\cal G}] onto [{\cal G}\,'] such that for any pair of elements of [{\cal G}] the image of the product is equal to the product of the images.

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[link].

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 [{\cal G}] is mapped onto the same element of [{\cal G}\,']. This concept leads to the definition of homomorphism.

Definition 1.2.3.2.3. A mapping of a group [{\cal G}] onto a group [{\cal G}\,'] is called homomorphic, and [{\cal G}\,'] is called a homomorphic image of the group [{\cal G}], if for any pair of elements of [{\cal G}] the image of the product is equal to the product of the images and if any element of [{\cal G}\,'] is the image of at least one element of [{\cal G}]. The relation of [{\cal G}] and [{\cal G}\,'] is called a homomorphism. More formally: For the mapping [{\cal G}] onto [{\cal G}\,'], [\ispecialfonts({\sfi g}_j\,{\sfi g}_k)'={\sfi g}_j'\,{\sfi g}_k'] holds.

The formulation `mapping onto' implies that each element [\ispecialfonts{\sfi g}'\in{\cal G}\,'] occurs among the images of the elements [\ispecialfonts{\sfi g}\in{\cal G}] at least once.3

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

References

Flack, H. D., Wondratschek, H., Hahn, Th. & Abrahams, S. C. (2000). Symmetry elements in space groups and point groups. Addenda to two IUCr reports on the nomenclature of symmetry. Acta Cryst. A56, 96–98.
Hahn, Th. & Wondratschek, H. (1994). Symmetry of Crystals. Introduction to International Tables for Crystallography, Vol. A. Sofia: Heron Press.
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.








































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