International
Tables for Crystallography Volume A Space-group symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 |
International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 10-11
Section 1.1.8. Conjugation, normalizers^{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 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
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.