International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 211
doi: 10.1107/97809553602060000919 Chapter 1.1. A general introduction to groups^{a}Radboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands In this chapter, we introduce the fundamental concepts of group theory with the focus on those properties that are of particular importance for crystallography. Among other examples, the symmetry groups of an equilateral triangle and of the square are used throughout to illustrate the various concepts, whereas the actual application to crystallographic space groups will be found in later chapters. Starting from basic principles, we proceed to subgroups and the coset decomposition with respect to a subgroup. A particular type of subgroup is a normal subgroup. These are distinguished by the fact that the cosets with respect to such a subgroup can themselves be regarded as the elements of a group, called a factor group. These concepts have a very natural application to crystallographic space groups, since the translation subgroup is a normal subgroup and the corresponding factor group is precisely the point group of the space group. We then show how groups can be related by introducing homomorphisms, which are mappings between the groups that are compatible with the group operation. An important link between abstract groups and groups of symmetry operations is the notion of a group action. This formalizes the idea that group elements are applied to objects like points in space. In particular, objects that are mapped to each other by a group element are often regarded as equivalent and the subgroup of group elements that fix an object provides an important characterization of this object. Applied to crystallographic space groups acting on points in space, this gives rise to the concept of Wyckoff positions. We finally look at the notion of conjugacy and at normalizers, which provide important information on the intrinsic ambiguity in the symmetry description of crystal structures. 
In this chapter we give a general introduction to group theory, which provides the mathematical background for considering symmetry properties. Starting from basic principles, we discuss those properties of groups that are of particular interest in crystallography. To readers interested in a more elaborate treatment of the theoretical background, the standard textbooks by Armstrong (2010), Hill (1999) or Sternberg (2008) are recommended; an account from the perspective of crystallography can also be found in Müller (2013).
Crystal structures may be investigated and classified according to their symmetry properties. But in a strict sense, crystal structures in nature are never perfectly symmetric, due to impurities, structural imperfections and especially their finite extent. Therefore, symmetry considerations deal with idealized crystal structures that are free from impurities and structural imperfections and that extend infinitely in all directions. In the mathematical model of such an idealized crystal structure, the atoms are replaced by points in a threedimensional point space and this model will be called a crystal pattern.
A symmetry operation of a crystal pattern is a transformation of threedimensional space that preserves distances and angles and that leaves the crystal pattern as a whole unchanged. The symmetry of a crystal pattern is then understood as the collection of all symmetry operations of the pattern.
The following simple statements about the symmetry operations of a crystal pattern are almost selfevident:
These observations (together with the fact that leaving all points in their position is also a symmetry operation) show that the symmetry operations of a crystal pattern form an algebraic structure called a group.
Although groups occur in innumerable contexts, their basic properties are very simple and are captured by the following definition.
Definition. Let be a set of elements on which a binary operation is defined which assigns to each pair of elements the composition . Then , together with the binary operation , is called a group if the following hold:
In most cases, the composition of group elements is regarded as a product and is written as or even instead of . An exception is groups where the composition is addition, e.g. a group of translations. In such a case, the composition is more conveniently written as .
Examples
If a group contains finitely many elements, it is called a finite group and the number of its elements is called the order of the group, denoted by . A group with infinitely many elements is called an infinite group.
For a group element , its order is the smallest integer such that is the identity element. If there is no such integer, then is said to be of infinite order.
The group operation is not required to be commutative, i.e. in general one will have . However, a group in which for all is said to be a commutative or abelian group.
The inverse of the product of two group elements is the product of the inverses of the two elements in reversed order, i.e. .
A particularly simple type of groups is cyclic groups in which all elements are powers of a single element . A finite cyclic group of order n can be written as . For example, the rotations that are symmetry operations of an equilateral triangle constitute a cyclic group of order 3.
The group of integers (with addition as operation) is an example of an infinite cyclic group in which negative powers also have to be considered, i.e. where .
Groups of small order may be displayed by their multiplication table, which is a square table with rows and columns indexed by the group elements and where the intersection of the row labelled by and of the column labelled by is the product . It follows immediately from the invertibility of the group elements that each row and column of the multiplication table contains every group element precisely once.
Examples
The groups that are considered in crystallography do not consist of abstract elements but of symmetry operations with a geometric meaning. In the figures illustrating the groups and also in the symbols used for the group elements, this geometric nature is taken into account. For example, the fourfold rotation 4^{+} in the group 4mm is represented by the small black square placed at the rotation point and the reflection m_{10} by the line fixed by the reflection. To each crystallographic symmetry operation a geometric element is assigned which characterizes the type of the symmetry operation. The precise definition of the geometric elements for the different types of operations is given in Section 1.2.3 . For a rotation in threedimensional space the geometric element is the line along the rotation axis and for a reflection it is the plane fixed by the reflection. Different symmetry operations may share the same geometric element, but these operations are then closely related, such as rotations around the same line. One therefore introduces the notion of a symmetry element, which is a geometric element together with its associated symmetry operations. In the figures for the crystallographic groups, the symbols like the little black square or the lines actually represent these symmetry elements (and not just a symmetry operation or a geometric element).
It is clear that for larger groups the multiplication table becomes unwieldy to set up and use. Fortunately, for many purposes a full list of all products in the group is actually not required. A very economic alternative of describing a group is to give only a small subset of the group elements from which all other elements can be obtained by forming products.
Definition. A subset is called a set of generators for if every element of can be obtained as a finite product of elements from or their inverses. If is a set of generators for , one writes .
A group which has a finite generating set is said to be finitely generated.
Examples

Although one usually chooses generating sets with as few elements as possible, it is sometimes convenient to actually include some redundancy. For example, it may be useful to generate the symmetry group 4mm of the square by . The element 2 is redundant, since , but this generating set explicitly shows the different types of elements of order 2 in the group.
The group of symmetry operations of a crystal pattern may alter if the crystal undergoes a phase transition. Often, some symmetries are preserved, while others are lost, i.e. symmetry breaking takes place. The symmetry operations that are preserved form a subset of the original symmetry group which is itself a group. This gives rise to the concept of a subgroup.
Definition. A subset is called a subgroup of if its elements form a group by themselves. This is denoted by . If is a subgroup of , then is called a supergroup of . In order to be a subgroup, is required to contain the identity element of , to contain inverse elements and to be closed with respect to composition of elements. Thus, technically, every group is a subgroup of itself.
The subgroups of that are not equal to are called proper subgroups of . A proper subgroup of is called a maximal subgroup if it is not a proper subgroup of any proper subgroup of .
It is often convenient to specify a subgroup of by a set of generators. This is denoted by . The order of is not a priori obvious from the set of generators. For example, in the symmetry group 4mm of the square the pairs and both generate subgroups of order 4, whereas the pair generates the full group of order 8.
The subgroups of a group can be visualized in a subgroup diagram. In such a diagram the subgroups are arranged with subgroups of higher order above subgroups of lower order. Two subgroups are connected by a line if one is a maximal subgroup of the other. By following downward paths in this diagram, all group–subgroup relations in a group can be derived. Additional information is provided by connecting subgroups of the same order by a horizontal line if they are conjugate (see Section 1.1.7).
Examples

A subgroup allows us to partition a group into disjoint subsets of the same size, called cosets.
Definition. Let be a subgroup of . Then for the set is called the left coset of with representative . Analogously, the right coset with representative is defined as The coset is called the trivial coset of .
Remarks

These two remarks have an important consequence: since an element is contained in the coset , the cosets of partition the elements of into sets of the same cardinality as (which is of the order of in the case where this is finite).
Definition. If the number of different cosets of a subgroup is finite, this number is called the index of in , denoted by or . Otherwise, is said to have infinite index in .
In the case of a finite group, the partitioning of the elements of into the cosets of shows that both the order of and the index of in divide the order of . This is summarized in the following famous result.
Lagrange's theorem
For a finite group and a subgroup of one has i.e. the order of a subgroup multiplied by its index gives the order of the full group.
For example, a group of order n cannot have a proper subgroup of order larger than .
Whether or not two cosets of a subgroup are equal depends on whether the quotient of their representatives is contained in : for left cosets one has if and only if and for right cosets if and only if .
Definition. If is a subgroup of and are such that for , and every is contained in some left coset , then is called a system of left coset representatives of relative to . It is customary to choose so that the coset is the subgroup itself. The decomposition is called the coset decomposition of into left cosets relative to .
Analogously, is called a system of right coset representatives if for and every is contained in some right coset . Again, one usually chooses and the decomposition is called the coset decomposition of into right cosets relative to .
To obtain the coset decomposition one starts by choosing as the first coset (with representative ). Next, an element with is selected as representative for the second coset . For the third coset, an element with and is required. If at a certain stage the cosets have been defined but do not yet exhaust , an element not contained in the union is chosen as representative for the next coset.
Examples

In general, the left and right cosets of a subgroup differ, for example in the symmetry group 3m of an equilateral triangle the left coset decomposition with respect to the subgroup is whereas the right coset decomposition is For particular subgroups, however, it turns out that the left and right cosets coincide, i.e. one has for all . This means that for every and every the element is of the form for some and thus . The element is called the conjugate of by . Note that in the definition of the conjugate element there is a choice whether the inverse element is placed to the left or right of . Depending on the applications that are envisaged and on the preferences of the author, both versions and are found in the literature, but in the context of crystallographic groups it is more convenient to have the inverse to the right of .
An important aspect of conjugate elements is that they share many properties, such as the order or the type of symmetry operation. As a consequence, conjugate symmetry operations have the same type of geometric elements. For example, if is a threefold rotation in threedimensional space, its geometric element is the line along the rotation axis. The geometric element of a conjugate element is then also a line fixed by a threefold rotation, but in general this line has a different direction.
Definition. A subgroup of is called a normal subgroup if for all and all . This is denoted by . For a normal subgroup , the left and right cosets of with respect to coincide.
Remarks

Examples

For a subgroup of and an element , the conjugates form a subgroup because . This subgroup is called the conjugate subgroup of by . As already noted, conjugation does not alter the type of symmetry operations and their geometric elements, but it is possible that the orientations of the geometric elements are changed.
Using the concept of conjugate subgroups, a normal subgroup is a subgroup that coincides with all its conjugate subgroups . This means that the set of geometric elements of a normal subgroup is not changed by conjugation; the single geometric elements may, however, be permuted by the conjugating element. In the example of the symmetry group 4mm discussed above, the normal subgroup contains the reflections m_{10} and m_{01} with the lines along the coordinate axes as geometric elements. These two lines are interchanged by the fourfold rotation 4^{+}, corresponding to the fact that conjugation by 4^{+} interchanges m_{10} and m_{01}. The concept of conjugation will be discussed in more detail in Section 1.1.8.
One of the main motivations for studying normal subgroups is that they allow us to define a group operation on the cosets of in . The products of any element in the coset with any element in the coset lie in a single coset, namely in the coset . Thus we can define the product of the two cosets and as the coset with representative .
Definition. The set together with the binary operation forms a group, called the factor group or quotient group of by .
The identity element of the factor group is the coset and the inverse element of is the coset .
A familiar example of a factor group is provided by the times on a clock. If it is 8 o'clock (in the morning) now, then we say that in nine hours it will be 5 o'clock (in the afternoon). We regard times as elements of the factor group in which = = . In the factor group , the clock is imagined as a circle of circumference 12 around which the line of integers is wrapped so that integers with a difference of 12 are located at the same position on the circle.
The clock example is a special case of factor groups of the integers. We have already seen that the set of multiples of a natural number n forms a subgroup of index n in . This is a normal subgroup, since is an abelian group. The factor group represents the addition of integers modulo n.
Examples

In order to relate two groups, mappings between the groups that are compatible with the group operations are very useful.
Recall that a mapping from a set A to a set B associates to each an element , denoted by and called the image of a (under ).
Definition. For two groups and , a mapping from to is called a group homomorphism or homomorphism for short, if it is compatible with the group operations in and , i.e. if The compatibility with the group operation is captured in the phrase
The image of the product is equal to the product of the images.
Fig. 1.1.6.1 gives a schematic description of the definition of a homomorphism. For to be a homomorphism, the two curved arrows are required to give the same result, i.e. first multiplying two elements in and then mapping the product to must be the same as first mapping the elements to and then multiplying them.
It follows from the definition of a homomorphism that the identity element of must be mapped to the identity element of and that the inverse of an element must be mapped to the inverse of the image of , i.e. that . In general, however, other elements than the identity element may also be mapped to the identity element of .
In the case where only the identity element of lies in the kernel of , one can conclude that implies and is called an injective homomorphism. In this situation no information about the group is lost and the homomorphism can be regarded as an embedding of into .
The image of any homomorphism from to forms not just a subset, but a subgroup of . It is not required that is all of , but if this happens to be the case, is called a surjective homomorphism.
Examples

Definition. A homomorphism from to is called an isomorphism if and , i.e. if is both injective and surjective. An isomorphism is thus a onetoone mapping between the elements of and which is also a homomorphism.
Groups and between which an isomorphism exist are called isomorphic groups, this is denoted by .
Isomorphic groups may differ in the way they are realized, but they coincide in their structure. In essence, one can regard isomorphic groups as the same group with different names or labels for the group elements. For example, isomorphic groups have the same multiplication table if the elements are relabelled according to the isomorphism identifying the elements of the first group with those of the second. If one wants to stress that a certain property of a group will be the same for all groups which are isomorphic to , one speaks of as an abstract group.
Examples

The kernel of a homomorphism is always a normal subgroup, since for and one has . The information about the elements in the kernel of is lost after applying , because they are all mapped to the identity element of . More precisely, if , then all elements from the coset are mapped to the same element in , since for one has . Conversely, if elements are mapped to the same element, they have to lie in the same coset, since implies , thus and thus , i.e. . The cosets of therefore partition the elements of according to their images under . This observation is summarized in the following result, which is one of the most powerful theorems in group theory.
Homomorphism theorem
Let be a homomorphism from to with kernel . Then the factor group is isomorphic to the image via the isomorphism .
Examples

The concept of a group is the essence of an abstraction process which distils the common features of various examples of groups. On the other hand, although abstract groups are important and interesting objects in their own right, they are particularly useful because the group elements act on something, i.e. they can be applied to certain objects. For example, symmetry groups act on the points in space, but they also act on lines or planes. Groups of permutations act on the symbols themselves, but also on ordered and unordered pairs. Groups of matrices act on the vectors of a vector space, but also on the subspaces. All these different actions can be described in a uniform manner and common concepts can be developed.
Definition. A group action of a group on a set assigns to each pair an object of such that the following hold:

One says that the object is moved to by .
Example
The abstract group occurs as symmetry group in threedimensional space with three different actions of :
Often, two objects and are regarded as equivalent if there is a group element moving to . This notion of equivalence is in fact an equivalence relation in the strict mathematical sense:

Via this equivalence relation, the action of partitions the objects in into equivalence classes, where the equivalence class of an object consists of all objects which are equivalent to .
Definition. Two objects lie in the same orbit under if there exists such that .
The set of all objects in the orbit of is called the orbit of under .
The set of group elements that do not move the object is a subgroup of called the stabilizer of in .
If the orbit of a group action is finite, the length of the orbit is equal to the index of the stabilizer and thus in particular a divisor of the group order (in the case of a finite group). Actually, the objects in an orbit are in a very explicit onetoone correspondence with the cosets relative to the stabilizer, as is summarized in the orbit–stabilizer theorem.
Orbit–stabilizer theorem
For a group acting on a set let be an object in and let be the stabilizer of in .

Example
The symmetry group of the square acts on the corners of a square as displayed in Fig. 1.1.7.1. All four points lie in a single orbit under and the stabilizer of the point 1 is , i.e. a subgroup of index 4, as required by the orbit–stabilizer theorem. The stabilizers of the other points are conjugate to : The stabilizer of corner 3 equals and the stabilizer of both the corners 2 and 4 is , which is conjugate to by the fourfold rotation 4^{+} which moves corner 1 to corner 2.
An ndimensional space group acts on the points of the ndimensional space . The stabilizer of a point is called the sitesymmetry group of P (in ). These sitesymmetry groups play a crucial role in the classification of positions in crystal structures. If the sitesymmetry group of a point P consists only of the identity element of , P is called a point in general position, points with nontrivial sitesymmetry groups are called points in special position.
According to the orbit–stabilizer theorem, points that are in the same orbit under the space group and which are thus symmetry equivalent have sitesymmetry groups that are conjugate subgroups of . This gives rise to the concept of Wyckoff positions: points with sitesymmetry groups that are conjugate subgroups of belong to the same Wyckoff position. As a consequence, points in the same orbit under certainly belong to the same Wyckoff position, but points may have the same sitesymmetry group without being symmetry equivalent. The Wyckoff position of a point P consists of the union of the orbits of all points Q that have the same sitesymmetry group as P. For a detailed discussion of the crucial notion of Wyckoff positions we refer to Section 1.4.4 .
Example
In the symmetry group 4mm of the square the points lying on the geometric element of (i.e. the reflection line) are clearly stabilized by . The origin has the full group 4mm as its sitesymmetry group, for all other points with the sitesymmetry group is the group generated by the reflection .
The orbit of a point with is the four points , where both and have sitesymmetry group and and have the conjugate sitesymmetry group . This means that the Wyckoff position of e.g. the point consists of the set of all points and with arbitrary , i.e. of the union of the geometric elements of and with the exception of their intersection . A complete description of the distribution of points among the Wyckoff positions of the group 4mm is given in Table 3.2.3.1 .
In this section we focus on two group actions which are of particular importance for describing intrinsic properties of a group, namely the conjugation of group elements and the conjugation of subgroups. These actions were mentioned earlier in Section 1.1.5 when we introduced normal subgroups.
A group acts on its elements via , i.e. by conjugation. Note that the inverse element is required on the righthand side of in order to fulfil the rule for a group action.
The orbits for this action are called the conjugacy classes of elements of or simply conjugacy classes of ; the conjugacy class of an element consists of all its conjugates with running over all elements of . Elements in one conjugacy class have e.g. the same order, and in the case of groups of symmetry operations they also share geometric properties such as being a reflection, rotation or rotoinversion. In particular, conjugate elements have the same type of geometric element.
The connection between conjugate symmetry operations and their geometric elements is even more explicit by the orbit–stabilizer theorem: If and are conjugate by , i.e. , then maps the geometric element of to the geometric element of .
Example
The rotation group of a cube contains six fourfold rotations and if the cube is in standard orientation with the origin in its centre, the fourfold rotations , and and their inverses have the lines along the coordinate axesas their geometric elements, respectively. The twofold rotation around the linemaps the a axis to the b axis and vice versa, therefore the symmetry operation conjugates to a fourfold rotation with the line along the b axis as geometric element. Since the positive part of the a axis is mapped to the positive part of the b axis and conjugation also preserves the handedness of a rotation, is conjugated to and not to the inverse element . The line along the c axis is fixed by , but its orientation is reversed, i.e. the positive and negative parts of the c axis are interchanged. Therefore, is conjugated to its inverse by .
For the conjugation action, the stabilizer of an element is called the centralizer of in , consisting of all elements in that commute with , i.e. .
Elements that form a conjugacy class on their own commute with all elements of and thus have the full group as their centralizer. The collection of all these elements forms a normal subgroup of which is called the centre of .
A group acts on its subgroups via = , i.e. by conjugating all elements of the subgroup. The orbits are called conjugacy classes of subgroups of . Considering the conjugation action of on its subgroups is often convenient, because conjugate subgroups are in particular isomorphic: an isomorphism from to is provided by the mapping .
The stabilizer of a subgroup of under this conjugation action is called the normalizer of in . The normalizer of a subgroup of is the largest subgroup of such that is a normal subgroup of . In particular, a subgroup is a normal subgroup of if and only if its normalizer is the full group .
The number of conjugate subgroups of in is equal to the index of in . According to the orbit–stabilizer theorem, the different conjugate subgroups of are obtained by conjugating with coset representatives for the cosets of relative to .
Examples

In the context of crystallographic groups, conjugate subgroups are not only isomorphic, but have the same types of geometric elements, possibly with different directions. In many situations it is therefore sufficient to restrict attention to representatives of the conjugacy classes of subgroups. Furthermore, conjugation with elements from the normalizer of a group permutes the geometric elements of the symmetry operations of . The role of the normalizer may in this situation be expressed by the phrase
The normalizer describes the symmetry of the symmetries.
Thus, the normalizer reflects an intrinsic ambiguity between different but equivalent descriptions of an object by its symmetries.
Example
The subgroup is a normal subgroup of the symmetry group of the square, and thus is the normalizer of in . As can be seen in the diagram in Fig. 1.1.7.1, the fourfold rotation 4^{+} maps the geometric element of the reflection m_{10} to the geometric element of m_{01} and vice versa, and fixes the geometric element of the rotation 4^{+}. Consequently, conjugation by 4^{+} fixes as a set, but interchanges the reflections m_{10} and m_{01}. These two reflections are geometrically indistinguishable, since their geometric elements are both lines through the centres of opposite edges of the square.
Analogously, 4^{+} interchanges the geometric elements of the reflections and of the subgroup . These are the two reflection lines through opposite corners of the square.
In contrast to that, does not contain an element mapping the geometric element of m_{10} to that of . Note that an eightfold rotation would be such an element, but this is, however, not a symmetry of the square. The reflections m_{10} and are thus geometrically different symmetry operations of the square.
References
Armstrong, M. A. (2010). Groups and Symmetry. New York: Springer.Hill, V. E. (1999). Groups and Characters. Boca Raton: Chapman & Hall/CRC.
Müller, U. (2013). Symmetry Relationships between Crystal Structures. Oxford: IUCr/Oxford University Press.
Sternberg, S. (2008). Group Theory and Physics. Cambridge University Press.