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

International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 2-11
doi: 10.1107/97809553602060000919

Chapter 1.1. A general introduction to groups

B. Souvigniera*

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

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).

1.1.1. Introduction

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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 three-dimensional point space and this model will be called a crystal pattern.

A symmetry operation of a crystal pattern is a transformation of three-dimensional 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 self-evident:

 (a) If two symmetry operations are applied successively, the crystal pattern is still invariant, thus the combination of the two operations (called their composition) is again a symmetry operation. (b) Every symmetry operation can be reversed by simply moving every point back to its original position.

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.

1.1.2. Basic properties of groups

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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:

 (i) the binary operation is associative, i.e. ; (ii) there exists a unit element or identity element such that and for all ; (iii) every has an inverse element, denoted by , for which .

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

 (i) The group consisting only of the identity element (with ) is called the trivial group. (ii) The group 3m of all symmetries of an equilateral triangle is a group with the composition of symmetry operations as binary operation. The group contains six elements, namely three reflections, two rotations and the identity element. It is schematically displayed in Fig. 1.1.2.2. (iii) The set of all integers forms a group with addition as operation. The identity element is 0, the inverse element for is . (iv) The set of complex numbers with absolute value 1 forms a circle in the complex plane, the unit circle . The unit circle can be described by and forms a group with (complex) multiplication as operation. (v) The set of all real matrices with determinant is a group with matrix multiplication as operation. This group is called the general linear group and denoted by .

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

• (i) A cyclic group of order 3 consists of the elements . Its multiplication table is

• (ii) The symmetry group 2mm of a rectangle (with unequal sides) consists of a twofold rotation 2, two reflections with mirror lines along the coordinate axes and the identity element 1 (see Fig. 1.1.2.1; the small black lenticular symbol in the centre represents the twofold rotation point).

 Figure 1.1.2.1 | top | pdf |Symmetry group 2mm of a rectangle.

Note that in this and all subsequent examples of crystallographic point groups we will use the Seitz symbols (cf. Section 1.4.2.2 ) for the symmetry operations and the Hermann–Mauguin symbols (cf. Section 1.4.1 ) for the point groups.

The multiplication table of the group 2mm is

 1 2 m10 m01 1 1 2 m10 m01 2 2 1 m01 m10 m10 m10 m01 1 2 m01 m01 m10 2 1

The symmetry of the multiplication table (with respect to the main diagonal) shows that this is an abelian group.

• (iii) The symmetry group 3m of an equilateral triangle consists (apart from the identity element 1) of the threefold rotations 3+ and 3 and the reflections m10, m01, m11 with mirror lines through a corner of the triangle and the centre of the opposite side (see Fig. 1.1.2.2; the small black triangle in the centre represents the threefold rotation point).

 Figure 1.1.2.2 | top | pdf |Symmetry group 3m of an equilateral triangle.

The multiplication table of the group 3m is

 1 3+ 3− m10 m01 m11 1 1 3+ 3− m10 m01 m11 3+ 3+ 3− 1 m11 m10 m01 3− 3− 1 3+ m01 m11 m10 m10 m10 m01 m11 1 3+ 3− m01 m01 m11 m10 3− 1 3+ m11 m11 m10 m01 3+ 3− 1

The fact that , but shows that this group is not abelian. It is actually the smallest group (in terms of order) that is not abelian.

• (iv) The symmetry group 4mm of the square consists of the cyclic group generated by the fourfold rotation 4+ containing the elements 1, 4+, 2, 4 and the reflections m10, m01, m11, with mirror lines along the coordinate axes and the diagonals of the square (see Fig. 1.1.2.3; the small black square in the centre represents the fourfold rotation point).

 Figure 1.1.2.3 | top | pdf |Symmetry group 4mm of the square.

The multiplication table of the group 4mm is

 1 2 4+ 4− m10 m01 m11 1 1 2 4+ 4− m10 m01 m11 2 2 1 4− 4+ m01 m10 m11 4+ 4+ 4− 2 1 m11 m01 m10 4− 4− 4+ 1 2 m11 m10 m01 m10 m10 m01 m11 1 2 4− 4+ m01 m01 m10 m11 2 1 4+ 4− m11 m11 m10 m01 4+ 4− 1 2 m11 m01 m10 4− 4+ 2 1

This group is not abelian, because for example , but .

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 m10 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 three-dimensional 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

 (i) Every finite group is finitely generated, since is allowed to consist of all group elements. (ii) A cyclic group is generated by a single element. In particular, the infinite cyclic group is generated by , but also by . (iii) The symmetry group 4mm of the square is generated by a fourfold rotation and any of the reflections, e.g. by , but also by two reflections with reflection lines which are not perpendicular, e.g. by . (iv) The full symmetry group of the cube consists of 48 elements. It can be generated by a fourfold rotation around the a axis, a threefold rotation around a space diagonal and the inversion . It is also possible to generate the group by only two elements, e.g. by the fourfold rotation and a reflection in a plane with normal vector along one of the face diagonals of the cube. (v) The additive group of the rational numbers is not finitely generated, because finite sums of finitely many generators have denominators dividing and thus is not a finite sum of these generators.

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.

1.1.3. Subgroups

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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

• (i) The set consisting only of the identity element of is a subgroup, called the trivial subgroup of .

• (ii) For the group of the integers, all subgroups are cyclic and generated by some integer n, i.e. they are of the form for an integer n. Such a subgroup is maximal if n is a prime number.

• (iii) For every element of a group , the powers of form a subgroup of which is a cyclic group.

• (iv) In the matrices of determinant 1 form a subgroup, since the determinant of the matrix product is equal to the product of the determinants of and .

• (v) In the symmetry group 3m of an equilateral triangle the rotations form a subgroup of order 3 (see Fig. 1.1.3.1).

 Figure 1.1.3.1 | top | pdf |Subgroup diagram for the symmetry group 3m of an equilateral triangle.
• (vi) The symmetry group 2mm of a rectangle has three subgroups of order 2, generated by the reflection m10, the twofold rotation 2 and the reflection m01, respectively (see Fig. 1.1.3.2).

 Figure 1.1.3.2 | top | pdf |Subgroup diagram for the symmetry group 2mm of a rectangle.
• (vii) In the symmetry group 4mm of the square, the reflections m10 and m01 together with their product 2 and the identity element 1 form a subgroup of order 4. This subgroup can be recognized in the subgroup diagram of 4mm as the subdiagram of the subgroups of in the left part of Fig. 1.1.3.3 which coincides with the subgroup diagram of 2mm in Fig. 1.1.3.2. A different subgroup of order 4 is formed by the other pair of perpendicular reflections , together with 2 and 1 and a third subgroup of order 4 is the cyclic subgroup generated by the fourfold rotation (see Fig. 1.1.3.3).

 Figure 1.1.3.3 | top | pdf |Subgroup diagram for the symmetry group 4mm of the square.

1.1.4. Cosets

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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

 (i) Since two elements and in the same coset can only be the same if , the elements of are in one-to-one correspondence with the elements of . In particular, for a finite subgroup the number of elements in each coset of equals the order of the subgroup . (ii) Every element contained in may serve as representative for this coset, i.e. for every . In particular, if an element is contained in the intersection of two cosets, one has and . This implies that two cosets are either disjoint (i.e. contain no common element) or they are equal.

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

 (i) Let be the symmetry group of an equilateral triangle and its subgroup containing the rotations. Then for every reflection the elements form a system of coset representatives of relative to and the coset decomposition is . (ii) For any integer n, the set of multiples of n forms an infinite subgroup of index n in . A system of coset representatives of relative to is formed by the numbers . The coset with representative 0 is , the coset with representative 1 is and an integer a belongs to the coset with representative k if and only if a gives remainder k upon division by n.

1.1.5. Normal subgroups, factor groups

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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 three-dimensional 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

 (i) The full group and the trivial subgroup are always normal subgroups of . These are often called the trivial normal subgroups of . (ii) In abelian groups, every subgroup is a normal subgroup, because implies . (iii) A subgroup of index 2 in is always a normal subgroup, since the coset decomposition relative to consists of only two cosets and for any element the left and right cosets and both consist precisely of those elements of that are not contained in . Therefore, for and for clearly holds. (iv) In order to check whether a subgroup of is a normal subgroup it is sufficient to check whether for generators of and generators of . This is due to the fact that on the one hand and on the other hand .

Examples

 (i) In the symmetry group 3m of an equilateral triangle, the subgroup generated by the threefold rotation 3+ is a normal subgroup because it is of index 2 in 3m. The subgroups of order 2 generated by the reflections m10, m01 and m11 are not normal because , and = . (ii) In the symmetry group 4mm of the square, the subgroups , , and are normal subgroups because they are subgroups of index 2. The subgroups of order 2 generated by the reflections m10, m01, and are not normal because , , = and = . The subgroup of order 2 generated by the twofold rotation 2 is normal because and .

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 m10 and m01 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 m10 and m01. 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

• (i) If we take to be the symmetry group 4mm of the square and choose as normal subgroup the subgroup generated by the fourfold rotation, we obtain a factor group with two elements, namely the cosets and . The trivial coset is the identity element in the factor group and contains the rotations in 4mm. The other element in the factor group consists of the reflections in 4mm.

In this example, the separation of the rotations and reflections in 4mm into the two cosets and makes it easy to see that the product of two cosets is independent of the chosen representative of the coset: the product of two rotations is again a rotation, hence , the product of a rotation and a reflection is a reflection, hence , and finally the product of two reflections is a rotation, hence . The multiplication table of the factor group is thus

• (ii) The symmetry group of a square is the same as the symmetry group of an eightfold star, as shown in Fig. 1.1.5.1. If we regard the star as being built from four lines (two dotted and two dashed), then the twofold rotation does not move any of the lines, it only interchanges the points within each line (symmetric with respect to the centre). Regarding the lines as sets of points, the twofold rotation thus does not change anything. The effects of the different symmetry operations on the lines of the eightfold star are then precisely given by the factor group , where is the symmetry group 4mm of the square and is the normal subgroup generated by the twofold rotation 2: the cosets relative to are , , , , and these cosets collect together the elements of 4mm that have the same effect on the lines of the eightfold star. For example, both 4+ and interchange both the two dotted and the two dashed lines, m10 and m01 both interchange the two dashed lines but fix the two dotted lines and and both interchange the two dotted lines but fix the two dashed lines. Owing to the fact that is a normal subgroup, the product of elements from two cosets always lies in the same coset, independent of which elements are chosen from the two cosets. For example, the product of an element from the coset with an element of the coset always gives an element of the coset . Working out the products for all pairs of cosets, one obtains the following multiplication table for the factor group :

 Figure 1.1.5.1 | top | pdf |Symmetry group of an eightfold star.
• (iii) If one takes cosets with respect to a subgroup that is not normal, the products of elements from two cosets do not lie in a single coset. As we have seen, the left cosets of the group 3m of an equilateral triangle with respect to the non-normal subgroup are , and . Taking products from elements of the first and second coset, we get and , which are both in the second coset, but and , which are both in the third coset.

1.1.6. Homomorphisms, isomorphisms

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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.

 Figure 1.1.6.1 | top | pdf |Schematic description of a homomorphism.

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 .

Definition. Let be a group homomorphism from to .

 (i) The set of elements mapped to the identity element of is called the kernel of , denoted by . (ii) The set is called the image of under .

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

• (i) For the symmetry group 4mm of the square a homomorphism to a cyclic group of two elements is given by and , i.e. by mapping the rotations in 4mm to the identity element of and the reflections to the non-trivial element.

Since every element of is the image of some element of 4mm, is a surjective homomorphism, but it is not injective because the kernel consists of all rotations in 4mm and not only of the identity element.

• (ii) The cyclic group of order n is mapped into the (multiplicative) group of the unit circle in the complex plane by mapping to . As displayed in Fig. 1.1.6.2, the image of under this homomorphism are points on the unit circle which form the corners of a regular n-gon.

 Figure 1.1.6.2 | top | pdf |Cyclic group of order 6 embedded in the group of the unit circle.

This is an injective homomorphism because the smallest with is and in , thus by this homomorphism can be regarded as a subgroup of . It is clear that cannot be surjective, because is an infinite group and the image consists of only finitely many elements.

• (iii) For the additive group of integers and a cyclic group , for every integer q a homomorphism is defined by mapping to , which gives for . This is never an injective homomorphism, because is contained in the kernel of . Whether or not is surjective depends on whether is a generator of . This is the case if and only if n and q have no non-trivial common divisors.

Definition. A homomorphism from to is called an isomorphism if and , i.e. if is both injective and surjective. An isomorphism is thus a one-to-one 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

 (i) The symmetry group 3m of an equilateral triangle is isomorphic to the group of all permutations of . This can be seen as follows: labelling the corners of the triangle by , each element of 3m gives rise to a permutation of the labels and mapping an element to the corresponding permutation is a homomorphism. The only element fixing all three corners of the triangle is the identity element of 3m, thus the homomorphism is injective. On the other hand, the groups 3m and both have 6 elements, hence the homomorphism is also surjective, and thus it is an isomorphism. (ii) For the symmetry group of the square and its normal subgroup generated by the fourfold rotation, the factor group is isomorphic to a cyclic group of order 2. The trivial coset (containing the rotations in 4mm) corresponds to the identity element , the other coset (containing the reflections) corresponds to . (iii) The real numbers form a group with addition as operation and the positive real numbers form a group with multiplication as operation. The exponential mapping is a homomorphism from to because = . It is an injective homomorphisms because only for [which is the identity element in ] and it is a surjective homomorphism because for any there is an with , namely . The exponential mapping therefore provides an isomorphism from to .

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

 (i) The homomorphism from 4mm to sending the rotations in 4mm to and the reflections to has the group of rotations in 4mm as its kernel. The factor group has the cosets = and as its elements and the homomorphism theorem confirms that mapping to and to is an isomorphism from to . (ii) The homomorphism from the additive group of integers to the cyclic group mapping k to has as its kernel. Since is a surjective homomorphism, the homomorphism theorem states that the factor group is isomorphic to the cyclic group . The operation in the factor group is `addition modulo n'.

1.1.7. Group actions

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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:

 (i) applying two group elements and consecutively has the same effect as applying the product , i.e. (note that since the group elements act from the left on the objects in , the elements in a product of two (or more) group elements are applied right-to-left); (ii) applying the identity element of has no effect on , i.e. for all in .

One says that the object is moved to by .

Example

The abstract group occurs as symmetry group in three-dimensional space with three different actions of :

 (i) If is a reflection, then the points fixed by form a two-dimensional plane. (ii) If is a twofold rotation, then the fixed points of form a one-dimensional line. (iii) If is an inversion, then only a single point is fixed by .

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:

 (a) it is reflexive, i.e. is equivalent to itself: this is easily seen since ; (b) it is symmetric, i.e. if is equivalent to , then is also equivalent to : this holds since implies = ; (c) it is transitive, i.e. if is equivalent to and is equivalent to , then is equivalent to : this is true because and implies .

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 one-to-one 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 .

 (i) If is the coset decomposition of relative to , then the coset consists of precisely those elements of that move to . As a consequence, the full orbit of is already obtained by applying only the coset representatives to , i.e. and the number of cosets equals the length of the orbit. (ii) For objects in the same orbit under , the stabilizers are conjugate subgroups of (cf. Section 1.1.5). If , then , i.e. the stabilizer of is obtained by conjugating the stabilizer of by the element moving to .

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.

 Figure 1.1.7.1 | top | pdf |Stabilizers in the symmetry group 4mm of the square.

An n-dimensional space group acts on the points of the n-dimensional space . The stabilizer of a point is called the site-symmetry group of P (in ). These site-symmetry groups play a crucial role in the classification of positions in crystal structures. If the site-symmetry group of a point P consists only of the identity element of , P is called a point in general position, points with non-trivial site-symmetry 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 site-symmetry groups that are conjugate subgroups of . This gives rise to the concept of Wyckoff positions: points with site-symmetry 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 site-symmetry 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 site-symmetry 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 site-symmetry group, for all other points with the site-symmetry group is the group generated by the reflection .

The orbit of a point with is the four points , where both and have site-symmetry group and and have the conjugate site-symmetry 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 .

1.1.8. Conjugation, normalizers

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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 right-hand 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

 (i) In an abelian group , every element is only conjugate with itself, since for all in . Therefore each conjugacy class consists of just a single element. Also, every subgroup of an abelian group is a normal subgroup, thus its normalizer is itself and is only conjugate to itself. (ii) The conjugacy classes of the symmetry group 3m of an equilateral triangle are , and . The centralizer of m10 is just the group generated by m10, i.e. 1 and m10 are the only elements of 3m commuting with m10. Analogously, one sees that the centralizer for each element in 3m, except for . The subgroups , and are conjugate sub­groups (with conjugating elements 3+ and ). These subgroups coincide with their normalizers, since they have index 3 in the full group. (iii) The conjugacy classes of the symmetry group 4mm of a square are , , , and . Since 2 forms a conjugacy class on its own, this is an element in the centre of 4mm and its centralizer is the full group. The centralizer of m10 is , which is also the centralizer of m01 (note that m10 and m01 are reflections with normal vectors perpendicular to each other, and thus commute). Analogously, is the centralizer of both and . Finally, 4+ only commutes with the rotations in 4mm, therefore its centralizer is . The five subgroups of order 2 in 4mm fall into three conjugacy classes, namely the normal subgroup and the two pairs and . The normalizer of both and is and the normalizer of both and is .

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 m10 to the geometric element of m01 and vice versa, and fixes the geometric element of the rotation 4+. Consequently, conjugation by 4+ fixes as a set, but interchanges the reflections m10 and m01. 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 m10 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 m10 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.