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

International Tables for Crystallography (2016). Vol. A, ch. 3.5, pp. 826-851
https://doi.org/10.1107/97809553602060000933

## Chapter 3.5. Normalizers of space groups and their use in crystallography

E. Koch,a* W. Fischera and U. Müllerb

aInstitut für Mineralogie, Petrologie und Kristallographie, Philipps-Universität, D-35032 Marburg, Germany, and bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:  kochelke@mailer.uni-marburg.de

In Section 3.5.1, the mathematical concept of Euclidean and affine normalizers of space groups is introduced. Some crystallographic problems are mentioned for which the solution of the problem is simplified by the use of normalizers. In Section 3.5.2, the properties of the Euclidean and affine normalizers of the plane groups and the space groups are discussed and described in detailed tables that also take into account the dependence of the Euclidean normalizers on the specialization of the metrical parameters for monoclinic and orthorhombic space groups. In addition, for the first time, chirality-preserving Euclidean normalizers of the space groups are listed. In Section 3.5.3, examples for the use of Euclidean and affine normalizers for crystallographic purposes are given: (i) The derivation of Euclidean- and affine-equivalent point configurations and Wyckoff positions constitutes the basis for the definition of Wyckoff sets. The derivation of all different coordinate descriptions of a certain crystal structure is described provided that the description of its space group (basis vectors and origin) remains unchanged. (ii) Each transition from one coordinate description of a crystal structure to another equivalent one necessarily causes changes in the corresponding list of structure factors: either the phases of the reflections or the phases and the indices are changed. As a consequence, the Euclidean normalizers of the space groups lead to a simple derivation of phase restrictions for use in direct methods to fix the origin and the enantiomorph'. (iii) Different subgroups (or supergroups) of a given space group that play an analogous role with respect to this space group may be identified with the aid of the Euclidean or affine normalizers. (iv) The ranges of the metrical and coordinate parameters that have to be considered for geometrical studies of point configurations can be reduced with the aid of the Euclidean and affine normalizers of space groups. Finally, in Section 3.5.4, the normalizers of the two-dimensional point groups with respect to the full isometry group of the circle and of the three-dimensional point groups with respect to the full isometry group of the sphere are tabulated.

### 3.5.1. Introduction and definitions

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E. Koch,a W. Fischera and U. Müllerb

#### 3.5.1.1. Introduction

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The mathematical concept of normalizers forms the common basis for the solution of several crystallographic problems:

It is generally known, for instance, that the coordinate description of a crystal structure trivially depends on the coordinate system used for the description, i.e. on the setting of the space group and the site symmetry of the origin. It is less well known, however, that for most crystal structures there exist several different but equivalent coordinate descriptions, even if the space-group setting and the site symmetry of the origin are unchanged. The number of such descriptions varies between 1 and 24 and depends only on the type of the Euclidean normalizer of the corresponding space group. In principle, none of these descriptions stands out against the others.

In crystal-structure determination with direct methods, the phases of some suitably chosen structure factors have to be restricted to certain values or to certain ranges in order to specify the origin and the enantiomorph. The information necessary for a correct selection of such phases and for their appropriate restrictions follows directly from the Euclidean normalizer of the space group. Similar examples are the positioning of the first atom(s) within an asymmetric unit when using trial-and-error or Patterson methods, the choice of a basis system for indexing the reflections of a diffraction pattern or the indexing of the first morphological face(s) of a crystal.

For the following problems, normalizers also play an important role: They supply information on the interchangeability of Wyckoff positions and their assignment to Wyckoff sets (cf. Section 1.4.4 and Chapter 3.4 ), needed e.g. for the definition of lattice complexes. They are important for the comparison of crystal structures, for their assignment to structure types and for the choice of a standard description for each crystal structure (Parthé & Gelato, 1984, 1985). They allow the derivation of privileged origins' for each space group (Burzlaff & Zimmermann, 1980) and facilitate the complete deduction of subgroups and supergroups of a crystallographic group. They enable an easy classification of magnetic (black–white or Shubnikov) space groups and of colour space groups. They may also be used to reduce the parameter range in the study of geometrical properties of point configurations, e.g. their eigensymmetry or their sphere packings and Dirichlet partitions (cf. e.g. Koch, 1984a).

In the past, most of these problems have been treated by crystallographers without the aid of normalizers, but the use of normalizers simplifies the solution of all these problems and clarifies the common background (for references, see Fischer & Koch, 1983).

#### 3.5.1.2. Definitions

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Any pair, consisting of a group and one of its supergroups , is uniquely related to a third intermediate group , called the normalizer of with respect to . is defined as the set of all elements that map onto itself by conjugation (cf. Section 1.1.8 ):The normalizer may coincide either with or with or it may be a proper intermediate group. In any case, is a normal subgroup of its normalizer.

For most crystallographic problems, three kinds of normalizers are of special interest:

 (i) The normalizer of a space group (plane group) with respect to the group of all Euclidean mappings (motions, isometries) in , called the Euclidean normalizer of : (ii) The normalizer of a space group (plane group) with respect to the group of all affine mappings in , called the affine normalizer of : (iii) The normalizer of a space group with respect to the group of all chirality-preserving Euclidean mappings in , i.e. of all translations and proper rotations (including screw rotations), but excluding symmetry operations of the second kind (viz. inversions, reflections, glide reflections and rotoinversions). We call it the chirality-preserving Euclidean normalizer of :

exists only if is a Sohncke space group. The 65 Sohncke space-group types are those space-group types that have no symmetry operations of the second kind (Flack, 2003).1 They include the eleven pairs of types of enantiomorphic space groups; these eleven pairs are the only ones where the space groups themselves are chiral, i.e. which have an Euclidean normalizer containing only isometries of the first kind. The space groups of the remaining 43 Sohncke types are not chiral but do allow chiral crystal structures. A rigid object (or spatial arrangement of points or atoms) is chiral if it is nonsuperposable by pure rotation or translation on its image formed by inversion through a point. A chiral crystal structure is compatible only with a Sohncke space group.

The Euclidean normalizers of the space groups were first derived by Hirshfeld (1968) under the name Cheshire groups. They have been tabulated in more detail by Gubler (1982a,b) and Fischer & Koch (1983). The Euclidean normalizers of triclinic and monoclinic space groups with specialized metric of the lattice were determined by Koch & Müller (1990). The affine normalizers of the space groups have been listed by Burzlaff & Zimmermann (1980), Billiet et al. (1982) and Gubler (1982a,b). They were also used for the derivation of Wyckoff sets and the definition of lattice complexes by Koch & Fischer (1975), even though there the automorphism groups of the space groups were tabulated instead of their affine normalizers. The chirality-preserving Euclidean normalizers are tabulated in this volume for the first time.

### 3.5.2. Euclidean and affine normalizers of plane groups and space groups

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E. Koch,a W. Fischera and U. Müllerb

#### 3.5.2.1. Euclidean normalizers of plane groups and space groups

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Since each symmetry operation of the Euclidean normalizer maps the space group onto itself, it also maps the set of all symmetry elements of onto itself. Therefore, the Euclidean normalizer of a space group can be interpreted as the group of motions that maps the pattern of symmetry elements of the space group onto itself, i.e. as the symmetry of the symmetry pattern'.

For most space (plane) groups, the Euclidean normalizers are space (plane) groups again. Exceptions are those groups where origins are not fully fixed by symmetry, i.e. all space groups of the geometrical crystal classes 1, m, 2, 2mm, 3, 3m, 4, 4mm, 6 and 6mm, and all plane groups of the geometrical crystal classes 1 and m. The Euclidean normalizer of each such group contains continuous translations (i.e. translations of infinitesimal length) in one, two or three independent lattice directions and, therefore, is not a space (plane) group but a supergroup of a space (plane) group.

If one regards a certain type of space (plane) group, usually the Euclidean normalizers of all corresponding groups belong also to only one type of normalizer. This is true for all cubic, hexagonal, trigonal and tetragonal space groups (hexagonal and square plane groups) and, in addition, for 21 types of orthorhombic space group (4 types of rectangular plane group), e.g. for Pnma.

In contrast to this, the Euclidean normalizer of a space (plane) group belonging to one of the other 38 orthorhombic (3 rectangular) types may interchange two or even three lattice directions if the corresponding basis vectors have equal length (example: Pmmm with a = b). Then, the Euclidean normalizer of this group belongs to the tetragonal (square) or even to the cubic crystal system, whereas another space (plane) group of the same type but with general metric has an orthorhombic (rectangular) Euclidean normalizer.

For each space (plane)-group type belonging to the monoclinic (oblique) or triclinic system, there also exist groups with specialized metric that have Euclidean normalizers of higher symmetry than for the general case (cf. Koch & Müller, 1990). The description of these special cases, however, is by far more complicated than for the orthorhombic system.

The symmetry of the Euclidean normalizer of a monoclinic (oblique) space (plane) group depends only on two metrical parameters. A clear presentation of all cases with specialized metric may be achieved by choosing the cosine of the monoclinic angle and the related axial ratio as parameters. To cover all different metrical situations exactly once, not all pairs of parameter values are allowed for a given type of space (plane) group, but one has to restrict the study to a certain parameter range depending on the type, the setting and the cell choice of the space (plane) group. Parthé & Gelato (1985) have discussed in detail such parameter regions for the first setting of the monoclinic space groups. Figs. 3.5.2.1 to 3.5.2.4 are based on these studies.

 Figure 3.5.2.1 | top | pdf |Parameter range for space groups of types and (plane groups of types p1 and p2). The information in parentheses refers to unique axis c.
 Figure 3.5.2.2 | top | pdf |Parameter range for space groups of types C2, Pc, Cm, Cc, , , and . They refer to the following settings: unique axis b, cell choice 2: P1n1, , ; unique axis b, cell choice 3: I121, I1m1, I1a1, , ; unique axis c, cell choice 2: P11n, , ; unique axis c, cell choice 3: I112, I11m, I11b, , . The information in parentheses refers to unique axis c.
 Figure 3.5.2.3 | top | pdf |Parameter range for space groups of types C2, Pc, Cm, Cc, , , and : unique axis b, cell choice 1: P1c1, , ; unique axis b, cell choice 2: A121, A1m1, A1n1, , ; unique axis c, cell choice 1: P11a, , ; unique axis c, cell choice 2: B112, B11m, B11n, , . The information in parentheses refers to unique axis c.
 Figure 3.5.2.4 | top | pdf |Parameter range for space groups of types C2, Pc, Cm, Cc, , , and : unique axis b, cell choice 1: C121, C1m1, C1c1, , ; unique axis b, cell choice 3: , , , ; unique axis c, cell choice 1: A112, A11m, A11a, , ; unique axis c, cell choice 3: P11b, , , . The information in parentheses refers to unique axis c.

Fig. 3.5.2.1 shows a suitably chosen parameter region for the five space-group types P2, , Pm, and and for the plane-group types p1 and p2. Each such space (plane) group with general metric may be uniquely assigned to an inner point of this region and any metrical specialization corresponds either to one of the three boundary lines or to one of their points of intersection and gives rise to a symmetry enhancement of the respective Euclidean normalizer.

For each of the other eight types of monoclinic space groups, i.e. C2, Pc, Cm, Cc, , , and , and for each setting three possibilities of cell choice are listed in Chapter 2.3 , which can be distinguished by different space-group symbols (example: , , , , , ). For each setting, there exist two ways to choose a suitable range for the metrical parameters such that each group corresponds to exactly one point:

 (i) One arbitrarily restricts oneself to cell choice 1, 2 or 3. Then, the suitable parameter range (displayed in one of the Figs. 3.5.2.2, 3.5.2.3 or 3.5.2.4) is larger than the range shown in Fig. 3.5.2.1 because, in contrast to the space-group types discussed above, some of the possible metrical specializations do not give rise to any symmetry enhancement of the Euclidean normalizers. These special metrical cases refer to the light lines subdividing the parameter regions of Figs. 3.5.2.2 to 3.5.2.4. Again, all inner points of these regions correspond to space groups with Euclidean normalizers without enhanced symmetry, and all points on the heavy-line boundaries refer to space groups, the Euclidean normalizers of which show symmetry enhancement. (ii) For all types of monoclinic space groups, one regards only the small parameter region shown in Fig. 3.5.2.1, but in return takes into consideration all three possibilities for the cell choice. Then, however, not all boundaries of this small parameter region correspond to Euclidean normalizers with enhanced symmetry. (Similar considerations are true for oblique plane groups.)

For triclinic space groups, five metrical parameters are necessary and, therefore, it is impossible to describe the special metrical cases in an analogous way.

In general, between a space group (or plane group) and its Euclidean normalizer , two uniquely defined intermediate groups and exist, such that holds. is that klassengleiche supergroup of that is at the same time a translationengleiche subgroup of . It is well defined according to the theorem of Hermann (1929). The group differs from only if is noncentrosymmetric but is centrosymmetric; then is that centrosymmetric supergroup of of index 2 that is again a subgroup of . It belongs to the Laue class of . If is noncentrosymmetric, an intermediate group cannot exist.

The chirality-preserving Euclidean normalizer of a Sohncke space group is the unique noncentrosymmetric sub­group of which is a supergroup of : If is centrosymmetric, is a subgroup of index 2 of . If is noncentrosymmetric, and are identical.

With the aid of its chirality-preserving Euclidean normalizer it is possible to determine all equivalent sets of coordinates of a chiral crystal structure, excluding the opposite enantiomorph (cf. Section 3.5.3.2).

The groups and are of special interest in connection with direct methods for structure determination: they cause the parity classes of reflections; defines the permissible origin shifts and the parameter ranges for the phase restrictions in the specification of the origin; and gives information on possible phase restrictions for the selection of the enantiomorph. For any space (plane) group , the translation subgroups of , , and even coincide.

The Euclidean normalizers of the plane groups are listed in Table 3.5.2.1, those of triclinic space groups in Table 3.5.2.2. The Euclidean and the chirality-preserving Euclidean normalizers of monoclinic and orthorhombic space groups are in Tables 3.5.2.3 and 3.5.2.4, those of all other space groups in Table 3.5.2.5. Herein all settings and choices of cell and origin as tabulated in Chapters 2.2 and 2.3 are taken into account and, in addition, all metrical specializations giving rise to Euclidean normalizers with enhanced symmetry. Each setting, cell choice, origin or metrical specialization corresponds to one line in the tables. (Exceptions are some orthorhombic space groups with tetragonal metric: if as well as and give rise to a symmetry enhancement of the Euclidean normalizer, only the case is listed in Table 3.5.2.4.)

 Table 3.5.2.1| top | pdf | Euclidean normalizers of the plane groups
 For the restrictions of the cell metric of the two oblique plane groups see text and Fig. 3.5.2.3.
Plane group Euclidean normalizer Additional generators of Index of in
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsTwofold rotationFurther generators
1 p1 General r, 0; 0, s
r, 0; 0, s
, r, 0; 0, s
, r, 0; 0, s y, x
r, 0; 0, s
r, 0; 0, s
2 p2 General p2
p2mm
, c2mm
, c2mm
p4mm
p6mm
3 p1m1
4 p1g1
5 c1m1
6 p2mm p2mm
p4mm   y, x
7 p2mg   p2mm
8 p2gg p2mm
p4mm   y, x
9 c2mm p2mm
p4mm   y, x
10 p4   p4mm   y, x
11 p4mm   p4mm
12 p4gm   p4mm
13 p3   p6mm , y, x
14 p3m1   p6mm ,
15 p31m   p6mm
16 p6   p6mm     y, x
17 p6mm   p6mm
 Table 3.5.2.2| top | pdf | Euclidean normalizers of the triclinic space groups
 Basis vectors of the Euclidean normalizers ( refer to the possibly centred conventional unit cell for the respective Bravais lattice): ; .
Bravais typeEuclidean normalizer of
P1 (1) (2)
aP
mP
mA
oP Pmmm
oC Cmmm
oF Fmmm
oI Immm
tP
tI
hP
hR
cP
cF
cI
 Table 3.5.2.3| top | pdf | Euclidean and chirality-preserving Euclidean normalizers of the monoclinic space groups
 For the restrictions of the cell metric see text and Figs. 3.5.2.1 to 3.5.2.4. The symbols in parentheses following a space-group symbol refer to the location of the origin (origin choice' in Chapter 2.3 ).
Space group Euclidean normalizer and chirality-preserving normalizer Additional generators of and Index of in or
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsInversion through a centre atFurther generators
3 General

3 General

4 General

4 General

5 General

5 General

5 General

5 General

5 General

5 General

6 General

6 General

7 General

7 General

7 General

7 General

7 General

7 General

8 General

8 General

8 General

8 General

8 General

8 General

9 General

9 General

9 General

9 General

9 General

9 General

10 General

10 General

11 General

11 General

12 General

12 General

12 General

12 General

12 General

12 General

13 General

13 General

13 General

13 General

13 General

13 General

14 General

14 General

14 General

14 General

14 General

14 General

15 General

15 General

15 General

15 General

15 General

15 General

 Table 3.5.2.4| top | pdf | Euclidean and chirality-preserving Euclidean normalizers of the orthorhombic space groups
 The symbols in parentheses following a space-group symbol refer to the location of the origin (origin choice' in Chapter 2.3 ).
Space group Euclidean normalizer and chirality-preserving normalizer Additional generators of and Index of in or
No.Hermann–Mauguin symbolCell metricSymbolBasis vectorsTranslationsInversion through a centre atFurther generators
16

17

18

19

20

21

22

23

24

25

26
27

28
29
30
31
32

33
34

35

36
37

38
39
40
41
42

43

44

45

46
47

48 (both
origins)

49

50 (both
origins)
51
52
53
54
55

56

57
58

59 (both
origins)
60
61 or or

62
63
64
65

66

67

68

68

69

70

70
( at )

71

72

73

74

 Table 3.5.2.5| top | pdf | Euclidean and chirality-preserving Euclidean normalizers of the tetragonal, trigonal, hexagonal and cubic space groups
 The symbols in parentheses following a space-group symbol refer to the location of the origin (origin choice' in Chapter 2.3 ).
Space group Euclidean normalizer and chirality-preserving normalizer Additional generators of and Index of in or
No.Hermann–Mauguin symbolSymbolBasis vectorsTranslationsInversion through a centre atFurther generators
75

76
77

78
79

80

81
82
83