International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. A1, ch. 1.5, pp. 34-36
Section 1.5.4. Space groups^{a}Abteilung Reine Mathematik, Universität Ulm, D-89069 Ulm, Germany |
In IT A (2005), Section 8.1.6 , space groups are introduced as symmetry groups of crystal patterns.
Definition 1.5.4.1.1
The definition introduced space groups in the way they occur in crystallography: The group of symmetries of an ideal crystal stabilizes the crystal structure. This definition is not very helpful in analysing the structure of space groups. If is a space group, then the translation subgroup is a normal subgroup of . It is even a characteristic subgroup of , hence fixed under every automorphism of . By Definition 1.5.4.1.1, its image under the inverse of the mapping in Example 1.5.3.4.4 defined by in is a full lattice . Since is an isomorphism from onto , the translation subgroup of is isomorphic to the lattice . In particular, one has and the subgroup , formed by the pth powers of elements in , is mapped onto . Lattices are well understood. Although they are infinite, they have a simple structure, so they can be examined algorithmically. Since they lie in a vector space, one can apply linear algebra to them.
Now we want to look at how this lattice fits into the space group . The affine group acts on by conjugation as well as on via its linear part. Similarly the space group acts on by conjugation: For and , one gets , where is the linear part of . Therefore the kernel of this action is on the one hand the centralizer of in , on the other hand, since contains a basis of , it is equal to the kernel of the mapping , which is , hence Hence only the linear part of acts faithfully on by conjugation and linearly on . This factor group is a finite group. Let us summarize this:
Theorem 1.5.4.1.2. Let be a space group. The translation subgroup is an Abelian normal subgroup of which is its own centralizer, . The finite group acts faithfully on by conjugation. This action is similar to the action of the linear part on the lattice .
Definition 1.5.4.2.1. A subgroup of a group is called maximal if and for all subgroups with it holds that either or .
The translation subgroup of the space group plays a very important role if one wants to analyse the space group . Let be a subgroup of . Then has either fewer translations () or the order of the linear part of , the index of in , gets smaller (), or both happen.
Definition 1.5.4.2.2. Let be a subgroup of the space group and .
Remark. The third isomorphism theorem, Theorem 1.5.3.5.2, implies that if is a k-subgroup, then . Hence is a k-subgroup if and only if .
Let be a maximal subgroup of . Then we have the following preliminary situation:
Since and , one has by Proposition 1.5.3.2.11 that . Hence the maximality of implies that or . If then , hence is a t-subgroup. If , then by the third isomorphism theorem, Theorem 1.5.3.5.2, , hence is a k-subgroup. This is given by the following theorem:
Theorem 1.5.4.2.3. (Hermann) Let be a maximal subgroup of the space group . Then is either a k-subgroup or a t-subgroup.
The above picture looks as follows in the two cases:
Let be a t-subgroup of . Then and is a subgroup of . On the other hand, any subgroup of defines a unique t-subgroup of with and , namely . Hence the t-subgroups of are in bijection to the subgroups of , which is a finite group according to the remarks below Definition 1.5.4.1.1. For future reference, we note this in the following corollary:
Corollary 1.5.4.2.4. The t-subgroups of the space group are in bijection with the subgroups of the finite group .
In the case , which is the most important case in crystallography, the finite groups are isomorphic to subgroups of either (Hermann–Mauguin symbol ) or (). Here denotes the direct product (cf. Definition 1.5.3.6.1), the cyclic group of order 2, and and the symmetric groups of degree 3 or 4, respectively (cf. Section 1.5.3.6). Hence the maximal subgroups of that are t-subgroups can be read off from the subgroups of the two groups above.
An algorithm for calculating the maximal t-subgroups of which applies to all three-dimensional space groups is explained in Section 1.5.5.
The more difficult task is the determination of the maximal k-subgroups.
Lemma 1.5.4.2.5. Let be a maximal k-subgroup of the space group . Then is a normal subgroup of . Hence is an -invariant lattice.
Proof. , so every element in can be written as where and . Therefore one obtains for since is Abelian. Since and is normal in , one has . But is a product of elements in and therefore lies in the subgroup , hence . QED
The candidates for translation subgroups of maximal k-subgroups of can be found by linear-algebra algorithms using the philosophy explained at the beginning of this section: acts on by conjugation and this action is isomorphic to the action of the linear part of on the lattice via the isomorphism . Normal subgroups of contained in are mapped onto -invariant sublattices of . An example for such a normal subgroup is the group formed by the pth powers of elements of for any natural number . One has .
If is a maximal k-subgroup of , then is a normal subgroup of that is maximal in , which means that is a maximal -invariant sublattice of . Hence it contains for some prime number p. One may view as a finite -module and find all candidates for such normal subgroups as full pre-images of maximal -submodules of . This gives an algorithm for calculating these normal subgroups, which is implemented in the package [CARAT].
The group is an Abelian group, with the additional property that for all one has . Such a group is called an elementary Abelian p-group.
From the reasoning above we find the following lemma.
Lemma 1.5.4.2.6. Let be a maximal k-subgroup of the space group . Then is an elementary Abelian p-group for some prime p. The order of is with .
Corollary 1.5.4.2.7. Maximal subgroups of space groups are again space groups and of finite index in the supergroup.
Hence the first step is the determination of subgroups of that are maximal in and normal in , and is solved by linear-algebra algorithms. These subgroups are the candidates for the translation subgroups for maximal k-subgroups . But even if one knows the isomorphism type of , the group does not in general determine . Given such a normal subgroup that is contained in , one now has to find all maximal k-subgroups with and . It might happen that there is no such group . This case does not occur if is a symmorphic space group in the sense of the following definition:
Definition 1.5.4.2.8. A space group is called symmorphic if there is a subgroup such that and . The subgroup is called a complement of the translation subgroup .
Note that the group in the definition is isomorphic to and hence a finite group.
If is symmorphic and is a complement of , then one may take .
This shows the following:
Lemma 1.5.4.2.9. Let be a symmorphic space group with translation subgroup and an -invariant subgroup of (i.e. ). Then there is at least one k-subgroup with translation subgroup .
In any case, the maximal k-subgroups, , of satisfy
To find these maximal subgroups, , one first chooses such a subgroup . It then suffices to compute in the finite group . If there is a complement of in , then every element may be written uniquely as with , . In particular, any other complement of in is of the form . One computes . Since is a subgroup of , it holds that . Moreover, every mapping with this property defines some maximal subgroup as above. Since and are finite, it is a finite problem to find all such mappings.
If there is no such complement , this means that there is no (maximal) k-subgroup of with .
References
International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)