International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 |
International Tables for Crystallography (2011). Vol. A1, ch. 1.4, pp. 39-40
Section 1.4.8. Minimal supergroups^{a}Lehrstuhl D für Mathematik, Rheinisch-Westfälische Technische Hochschule, D-52062 Aachen, Germany |
For several problems, for example for the prediction of a phase transition or in the search for overlooked symmetry in crystal-structure determinations etc., it is helpful to know all space groups containing a given space group , which means that . Then is called a supergroup of . Note that – in contrast to subgroups – the supergroups containing a space group of finite index need not be space groups. For instance, the one-dimensional translation grouphas a supergroup of index 2 isomorphic to which is not a subgroup of the one-dimensional affine group. is Abelian but has an element of finite order, so cannot be a space group. For the applications in crystallography, we are restricted to those supergroups of that are again space groups.
Clearly is a t-supergroup (or k-supergroup, respectively) of if and only if is a t-subgroup (or k-subgroup) of and the theorem of Hermann implies the following:
Theorem 1.4.8.2. (Theorem of Hermann.) Let and be space groups such that is a minimal supergroup of . Then is either a k-supergroup or a t-supergroup.
The determination of the k-supergroups of a given space group is the easier task. For instance, if is a symmorphic space group then all its k-supergroups are also symmorphic. This is not true for k-subgroups of .
Theorem 1.4.8.3. Let be a k-supergroup of the space group . Then . If is a minimal k-supergroup of then the translation lattice of is an -invariant lattice that contains as a maximal sublattice.
Proof. Let be the translation subgroup of . Then and is isomorphic to the point group , which is a finite group. By the isomorphism theorem Therefore, group generated by and is a subgroup of containing with the same index and, therefore, . Moreover, contains and hence contains . Since is a normal subgroup of , the space group acts on by conjugation and therefore is invariant. If there is an invariant lattice such that , then, applying the isomorphism μ from Example 1.4.3.4.4, the group is a space group with . Hence the minimality of the supergroup implies that is an -invariant lattice that contains as a maximal sublattice. QED
As for maximal k-subgroups, the index of a minimal k-supergroup of is a prime power and for each prime p there is some such that has a minimal k-supergroup of index . Because of the infinite number of prime numbers, a space group has infinitely many minimal k-supergroups, but there are only finitely many minimal k-supergroups containing a given space group of given index.
This is different for t-supergroups, as the following example shows.
Example 1.4.8.4
Letbe the two-dimensional translation group . Then for all the groupis a minimal t-supergroup of containing of index 2. These groups are conjugate under the normalizer of in the affine group [see Example 3.47 in Heidbüchel (2003)], Visually this means that the discrete sets of symmetry lines of the different plane groups may be shifted by any real distance against each other or relative to an arbitrarily chosen origin. This yields uncountably many different t-supergroups of which are all of the same type.
The use of the affine and Euclidean normalizers of a space group is described in Part 15 of IT A. The affine normalizer of an n-dimensional space group acts on the set of all minimal t-supergroups of by conjugation.
Proof. Let be representatives of the -orbits on the set of n-dimensional space groups, i.e. of the types of n-dimensional space groups. For letdenote the set of all (maximal) t-subgroups of that are isomorphic to .
If is a (minimal) t-supergroup of the space group , then there is some and such that and , hence the pair of space groups for some , and .
If is a second supergroup of and such that for the same , then normalizes . Hence there are at most orbits of on the set of (minimal) t-supergroups of . QED
This proof also provides an algorithm to determine representatives of the -orbits of minimal t-supergroups of a given space group , provided that one knows representatives of all affine classes of space groups and their maximal t-subgroups. For dimensions 2 and 3 these are given in this volume. Since maximal t-subgroups of three-dimensional space groups have index 2, 3 or 4, this also holds for the minimal t-supergroups of these groups.
Up to dimension , the minimal t-supergroups and the minimal k-supergroups of a given space group can be obtained with the commands TSupergroups and KSupergroups in CARAT [see also Heidbüchel (2003)].
References
Heidbüchel, O. (2003). Beiträge zur Theorie der kristallographischen Raumgruppen. Aachener Beiträge zur Mathematik, 29. ISBN 3–86073-649–3.