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. 10-11

Section 1.1.8. Conjugation, normalizers

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

1.1.8. Conjugation, normalizers

| top | pdf |

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[link] when we introduced normal subgroups.

A group [{\cal G}] acts on its elements via [\ispecialfonts{\sfi g}({\sfi h}): = {\sfi g} {\sfi h} {\sfi g}^{-1}], i.e. by conjugation. Note that the inverse element [\ispecialfonts{\sfi g}^{-1}] is required on the right-hand side of [\ispecialfonts{\sfi h}] in order to fulfil the rule [\ispecialfonts{\sfi g}({\sfi g}'({\sfi h})) = ({\sfi g} {\sfi g}')({\sfi h})] for a group action.

The orbits for this action are called the conjugacy classes of elements of [{\cal G}] or simply conjugacy classes of [{\cal G}]; the conjugacy class of an element [\ispecialfonts{\sfi h}] consists of all its conjugates [\ispecialfonts{\sfi g} {\sfi h} {\sfi g}^{-1}] with [\ispecialfonts{\sfi g}] running over all elements of [{\cal G}]. 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 [\ispecialfonts{\sfi h}] and [\ispecialfonts{\sfi h}'] are conjugate by [\ispecialfonts{\sfi g}], i.e. [\ispecialfonts{\sfi h}' = {\sfi g} {\sfi h} {\sfi g}^{-1}], then [\ispecialfonts{\sfi g}] maps the geometric element of [\ispecialfonts{\sfi h}] to the geometric element of [\ispecialfonts{\sfi h}'].

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 [{4}^+_{100}], [{4}^+_{010}] and [{4}^+_{001}] and their inverses have the lines along the coordinate axes[\left\{\pmatrix{x \cr 0 \cr 0 } \mid x \in {\bb R} \right\}, \left\{\pmatrix{0 \cr y \cr 0} \mid y \in {\bb R} \right\} \ {\rm and}\ \left\{ \pmatrix{0 \cr 0 \cr z} \mid z \in {\bb R} \right\}]as their geometric elements, respectively. The twofold rotation [{2}_{110}] around the line[\left\{ \pmatrix{x \cr x \cr 0 } \mid x \in {\bb R} \right\}]maps the a axis to the b axis and vice versa, therefore the symmetry operation [{2}_{110}] conjugates [{4}^+_{100}] 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, [{4}^+_{100}] is conjugated to [{4}^+_{010}] and not to the inverse element [{4}^-_{010}]. The line along the c axis is fixed by [{2}_{110}], but its orientation is reversed, i.e. the positive and negative parts of the c axis are interchanged. Therefore, [{4}^+_{001}] is conjugated to its inverse [{4}^-_{001}] by [{2}_{110}].

For the conjugation action, the stabilizer of an element [\ispecialfonts{\sfi h}] is called the centralizer [\ispecialfonts{\cal C_G({\sfi h})}] of [\ispecialfonts{\sfi h}] in [{\cal G}], consisting of all elements in [{\cal G}] that commute with [\ispecialfonts{\sfi h}], i.e. [\ispecialfonts{\cal C_G({\sfi h})} = \{ {\sfi g} \in {\cal G} \mid {\sfi g} {\sfi h} = {\sfi h} {\sfi g} \}].

Elements that form a conjugacy class on their own commute with all elements of [{\cal G}] and thus have the full group as their centralizer. The collection of all these elements forms a normal subgroup of [{\cal G}] which is called the centre of [{\cal G}].

A group [{\cal G}] acts on its subgroups via [\ispecialfonts{\sfi g}({\cal H}): =] [\ispecialfonts {\sfi g} {\cal H} {\sfi g}^{-1} ] = [\ispecialfonts \{ {\sfi g} {\sfi h} {\sfi g}^{-1} \mid {\sfi h} \in {\cal H} \}], i.e. by conjugating all elements of the subgroup. The orbits are called conjugacy classes of subgroups of [{\cal G}]. Considering the conjugation action of [{\cal G}] on its subgroups is often convenient, because conjugate subgroups are in particular isomorphic: an isomorphism from [{\cal H}] to [\ispecialfonts{\sfi g} {\cal H} {\sfi g}^{-1}] is provided by the mapping [\ispecialfonts{\sfi h}\, \mapsto\, {\sfi g} {\sfi h} {\sfi g}^{-1}].

The stabilizer of a subgroup [{\cal H}] of [{\cal G}] under this conjugation action is called the normalizer [{\cal N_G(H)}] of [{\cal H}] in [{\cal G}]. The normalizer of a subgroup [{\cal H}] of [{\cal G}] is the largest subgroup [{\cal N}] of [{\cal G}] such that [{\cal H}] is a normal subgroup of [{\cal N}]. In particular, a subgroup is a normal subgroup of [{\cal G}] if and only if its normalizer is the full group [{\cal G}].

The number of conjugate subgroups of [{\cal H}] in [{\cal G}] is equal to the index of [{\cal N_G(H)}] in [{\cal G}]. According to the orbit–stabilizer theorem, the different conjugate subgroups of [{\cal H}] are obtained by conjugating [{\cal H}] with coset representatives for the cosets of [{\cal G}] relative to [{\cal N_G(H)}].

Examples

  • (i) In an abelian group [{\cal G}], every element is only conjugate with itself, since [\ispecialfonts{\sfi g} {\sfi h} {\sfi g}^{-1} = {\sfi h}] for all [\ispecialfonts{\sfi g}, {\sfi h}] in [{\cal G}]. Therefore each conjugacy class consists of just a single element.

    Also, every subgroup [{\cal H}] of an abelian group [{\cal G}] is a normal subgroup, thus its normalizer [{\cal N_G(H)}] is [{\cal G}] itself and [{\cal H}] is only conjugate to itself.

  • (ii) The conjugacy classes of the symmetry group 3m of an equilateral triangle are [\{ {1} \}], [\{ {m}_{10}, {m}_{01}, {m}_{11} \}] and [\{ {3}^+, {3}^{-} \}]. The centralizer of m10 is just the group [\langle {m}_{10} \rangle] generated by m10, i.e. 1 and m10 are the only elements of 3m commuting with m10. Analogously, one sees that the centralizer [\ispecialfonts{\cal C_G}({\sfi h}) = \langle {\sfi h} \rangle] for each element [\ispecialfonts{\sfi h}] in 3m, except for [\ispecialfonts{\sfi h} = {1}].

    The subgroups [\langle {m}_{10} \rangle], [\langle {m}_{01} \rangle] and [\langle{m}_{11} \rangle] are conjugate sub­groups (with conjugating elements 3+ and [{3}^{-}]). 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 [\{ {1} \}], [\{ {2} \}], [\{{m}_{10}, {m}_{01} \}], [\{ {m}_{11}, {m}_{1\bar1} \}] and [\{ {4}^+, {4}^{-} \}]. 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 [\langle {m}_{10}, {m}_{01} \rangle], 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, [\langle {m}_{11}, {m}_{1\bar1} \rangle] is the centralizer of both [{m}_{11}] and [{m}_{1\bar1}]. Finally, 4+ only commutes with the rotations in 4mm, therefore its centralizer is [\langle {4}^+ \rangle].

    The five subgroups of order 2 in 4mm fall into three conjugacy classes, namely the normal subgroup [\langle {2} \rangle] and the two pairs [\{ \langle {m}_{10} \rangle, \langle {m}_{01} \rangle \}] and [\{ \langle {m}_{11} \rangle, \langle {m}_{1\bar1} \rangle \}]. The normalizer of both [\langle {m}_{10} \rangle] and [\langle{m}_{01} \rangle] is [\langle {2}, {m}_{10} \rangle] and the normalizer of both [\langle {m}_{11} \rangle] and [\langle {m}_{1\bar1} \rangle] is [\langle {2}, {m}_{11} \rangle].

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 [{\cal H}] permutes the geometric elements of the symmetry operations of [{\cal H}]. 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 [{\cal H} = \langle {2}, {m}_{10} \rangle] is a normal subgroup of the symmetry group [{\cal G} =4mm] of the square, and thus [{\cal G}] is the normalizer of [{\cal H}] in [{\cal G}]. As can be seen in the diagram in Fig. 1.1.7.1[link], 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 [{\cal H}] 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 [{m}_{11}] and [{m}_{1\bar1}] of the subgroup [{\cal H}' = \langle {2}, {m}_{11} \rangle]. These are the two reflection lines through opposite corners of the square.

In contrast to that, [{\cal G}] does not contain an element mapping the geometric element of m10 to that of [{m}_{11}]. Note that an eightfold rotation would be such an element, but this is, however, not a symmetry of the square. The reflections m10 and [m_{11}] are thus geometrically different symmetry operations of the square.








































to end of page
to top of page