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. 9-10

Section 1.1.7. Group actions

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

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 [{\cal G}] on a set [\Omega = \{ \omega \mid \omega \in \Omega \}] assigns to each pair [\ispecialfonts({\sfi g}, \omega)] an object [\ispecialfonts\omega' = {\sfi g}(\omega)] of [\Omega] such that the following hold:

  • (i) applying two group elements [\ispecialfonts{\sfi g}] and [\ispecialfonts{\sfi g}'] consecutively has the same effect as applying the product [\ispecialfonts{\sfi g}' {\sfi g}], i.e. [\ispecialfonts{\sfi g}'({\sfi g}(\omega)) =] [\ispecialfonts ({\sfi g}' {\sfi g})(\omega)] (note that since the group elements act from the left on the objects in [\Omega], the elements in a product of two (or more) group elements are applied right-to-left);

  • (ii) applying the identity element [\ispecialfonts{\sfi e}] of [{\cal G}] has no effect on [\omega], i.e. [\ispecialfonts{\sfi e}(\omega) = \omega] for all [\omega] in [\Omega].

One says that the object [\omega] is moved to [\ispecialfonts{\sfi g}(\omega)] by [\ispecialfonts{\sfi g}].

Example

The abstract group [\ispecialfonts{\cal C}_2 = \{ {\sfi e}, {\sfi g} \}] occurs as symmetry group in three-dimensional space with three different actions of [\ispecialfonts{\sfi g}]:

  • (i) If [\ispecialfonts{\sfi g}] is a reflection, then the points fixed by [\ispecialfonts{\sfi g}] form a two-dimensional plane.

  • (ii) If [\ispecialfonts{\sfi g}] is a twofold rotation, then the fixed points of [\ispecialfonts{\sfi g}] form a one-dimensional line.

  • (iii) If [\ispecialfonts{\sfi g}] is an inversion, then only a single point is fixed by [\ispecialfonts{\sfi g}].

Often, two objects [\omega] and [\omega'] are regarded as equivalent if there is a group element moving [\omega] to [\omega']. This notion of equivalence is in fact an equivalence relation in the strict mathematical sense:

  • (a) it is reflexive, i.e. [\omega] is equivalent to itself: this is easily seen since [\ispecialfonts{\sfi e}(\omega) = \omega];

  • (b) it is symmetric, i.e. if [\omega] is equivalent to [\omega'], then [\omega'] is also equivalent to [\omega]: this holds since [\ispecialfonts{\sfi g}(\omega) = \omega'] implies [\ispecialfonts{\sfi g}^{-1}(\omega') ] = [ \omega];

  • (c) it is transitive, i.e. if [\omega] is equivalent to [\omega'] and [\omega'] is equivalent to [\omega''], then [\omega] is equivalent to [\omega'']: this is true because [\ispecialfonts{\sfi g}(\omega) = \omega'] and [\ispecialfonts{\sfi g}'(\omega') = \omega''] implies [\ispecialfonts{\sfi g}' {\sfi g}(\omega) = \omega''].

Via this equivalence relation, the action of [{\cal G}] partitions the objects in [\Omega] into equivalence classes, where the equivalence class of an object [\omega \in \Omega] consists of all objects which are equivalent to [\omega].

Definition. Two objects [\omega, \omega' \in \Omega] lie in the same orbit under [{\cal G}] if there exists [\ispecialfonts{\sfi g} \in {\cal G}] such that [\ispecialfonts\omega' = {\sfi g}(\omega)].

The set [\ispecialfonts{\cal G}(\omega): = \{ {\sfi g}(\omega) \mid {\sfi g} \in {\cal G} \}] of all objects in the orbit of [\omega] is called the orbit of [\omega] under [{\cal G}].

The set [\ispecialfonts S_{{\cal G}}(\omega): = \{ {\sfi g} \in {\cal G} \mid {\sfi g}(\omega) = \omega \}] of group elements that do not move the object [\omega] is a subgroup of [{\cal G}] called the stabilizer of [\omega] in [{\cal G}].

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 [{\cal G}] acting on a set [\Omega] let [\omega] be an object in [\Omega] and let [S_{{\cal G}}(\omega)] be the stabilizer of [\omega] in [{\cal G}].

  • (i) If [\ispecialfonts{\sfi g_1} S_{{\cal G}}(\omega) \cup {\sfi g_2} S_{{\cal G}}(\omega) \cup \ldots \cup {\sfi g_m} S_{{\cal G}}(\omega)] is the coset decomposition of [{\cal G}] relative to [S_{{\cal G}}(\omega)], then the coset [\ispecialfonts{\sfi g_i} S_{{\cal G}}(\omega)] consists of precisely those elements of [{\cal G}] that move [\omega] to [\ispecialfonts{\sfi g_i}(\omega)]. As a consequence, the full orbit of [\omega] is already obtained by applying only the coset representatives to [\omega], i.e. [\ispecialfonts{\cal G}(\omega) =] [\ispecialfonts \{ {\sfi g_1}(\omega), {\sfi g_2}(\omega), \ldots, {\sfi g_m}(\omega) \}] and the number of cosets equals the length of the orbit.

  • (ii) For objects in the same orbit under [{\cal G}], the stabilizers are conjugate subgroups of [{\cal G}] (cf. Section 1.1.5[link]). If [\ispecialfonts\omega' = {\sfi g}(\omega)], then [\ispecialfonts S_{{\cal G}}(\omega') = {\sfi g} S_{{\cal G}}(\omega) {\sfi g}^{-1}], i.e. the stabilizer of [\omega'] is obtained by conjugating the stabilizer of [\omega] by the element [\ispecialfonts{\sfi g}] moving [\omega] to [\omega'].

Example

The symmetry group [{\cal G} = 4mm] of the square acts on the corners of a square as displayed in Fig. 1.1.7.1[link]. All four points lie in a single orbit under [{\cal G}] and the stabilizer of the point 1 is [{\cal H} = \langle {m}_{1\bar1} \rangle], i.e. a subgroup of index 4, as required by the orbit–stabilizer theorem. The stabilizers of the other points are conjugate to [{\cal H}]: The stabilizer of corner 3 equals [{\cal H}] and the stabilizer of both the corners 2 and 4 is [\langle {m}_{11} \rangle], which is conjugate to [{\cal H}] by the fourfold rotation 4+ which moves corner 1 to corner 2.

[Figure 1.1.7.1]

Figure 1.1.7.1 | top | pdf |

Stabilizers in the symmetry group 4mm of the square.

An n-dimensional space group [{\cal G}] acts on the points of the n-dimensional space [{\bb R}^n]. The stabilizer of a point [P \in {\bb R}^n] is called the site-symmetry group of P (in [{\cal G}]). 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 [{\cal G}], 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 [{\cal G}]. This gives rise to the concept of Wyckoff positions: points with site-symmetry groups that are conjugate subgroups of [{\cal G}] belong to the same Wyckoff position. As a consequence, points in the same orbit under [{\cal G}] 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[link] .

Example

In the symmetry group 4mm of the square the points [x,0] lying on the geometric element of [{m}_{01}] (i.e. the reflection line) are clearly stabilized by [{m}_{01}]. The origin [0,0] has the full group 4mm as its site-symmetry group, for all other points [x,0] with [x \neq 0] the site-symmetry group is the group [\langle{m}_{01} \rangle] generated by the reflection [{m}_{01}].

The orbit of a point [P = x, 0] with [x \neq 0] is the four points [x,0,\ \, 0,x,\ \, -x,0,\ \, 0,-x], where both [x,0] and [-x,0] have site-symmetry group [\langle {m}_{01}\rangle] and [0,x] and [0,-x] have the conjugate site-symmetry group [\langle {m}_{10} \rangle]. This means that the Wyckoff position of e.g. the point [P = {{1}\over{2}}, 0] consists of the set of all points [x,0] and [0,x] with arbitrary [x \neq 0], i.e. of the union of the geometric elements of [{m}_{01}] and [{m}_{10}] with the exception of their intersection [0,0]. A complete description of the distribution of points among the Wyckoff positions of the group 4mm is given in Table 3.2.3.1[link] .








































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