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. 1.2, pp. 22-23
Section 1.2.8. Lemmata on subgroups of space groups^{a}Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany |
There are several lemmata on subgroups of space groups which may help in getting an insight into the laws governing group–subgroup relations of plane and space groups. They were also used for the derivation and the checking of the tables of Part 2 . These lemmata are proved or at least stated and explained in Chapter 1.5 . They are repeated here as statements, separated from their mathematical background, and are formulated for the three-dimensional space groups. They are valid by analogy for the (two-dimensional) plane groups.
Lemma 1.2.8.1.1. A subgroup of a space group is a space group again, if and only if the index is finite.
In this volume, only subgroups of finite index i are listed. However, the index i is not restricted, i.e. there is no number I with the property for any i. Subgroups with infinite index are considered in International Tables for Crystallography, Vol. E (2002).
Lemma 1.2.8.1.2. Hermann's theorem. For any group–subgroup chain between space groups there exists a uniquely defined space group with , where is a translationengleiche subgroup of and is a klassengleiche subgroup of .
The decisive point is that any group–subgroup chain between space groups can be split into a translationengleiche subgroup chain between the space groups and and a klassengleiche subgroup chain between the space groups and .
It may happen that either or holds. In particular, one of these equations must hold if is a maximal subgroup of .
Lemma 1.2.8.1.3. (Corollary to Hermann's theorem.) A maximal subgroup of a space group is either a translationengleiche subgroup or a klassengleiche subgroup, never a general subgroup.
The following lemma holds for space groups but not for arbitrary groups of infinite order.
Lemma 1.2.8.1.4. For any space group, the number of subgroups with a given finite index i is finite.
This number of subgroups can be further specified, see Chapter 1.5 . Although for each index i the number of subgroups is finite, the number of all subgroups with finite index is infinite because there is no upper limit for the number i.
Even the set of all maximal subgroups of finite index is not finite, as can be seen from the following lemma.
Lemma 1.2.8.2.1. The index i of a maximal subgroup of a space group is always of the form , where p is a prime number and n =1 or 2 for plane groups and n = 1, 2 or 3 for space groups.
An index of occurs only for isomorphic subgroups of tetragonal, trigonal and hexagonal space groups when the basis vectors are enlarged to pa, pb. An index of occurs for and only for isomorphic subgroups of cubic space groups with cell enlargements of pa, pb, pc ().
This lemma means that a subgroup of, say, index 6 cannot be maximal. Moreover, because of the infinite number of primes, the set of all maximal subgroups of a given space group cannot be finite.
There are even stronger restrictions for maximal non-isomorphic subgroups.
Lemma 1.2.8.2.2. The index of a maximal non-isomorphic subgroup of a plane group is 2 or 3; for a space group the index is 2, 3 or 4.
This lemma can be specified further:
Lemma 1.2.8.2.3. The index of a maximal non-isomorphic subgroup is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups . The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups . The index is 2, 3 or 4 for cubic space groups .
There are also lemmata for the number of subgroups of a certain index. The most important are:
Lemma 1.2.8.2.4. The number of subgroups of index 2 is with for space groups and for plane groups. The number of translationengleiche subgroups of index 2 is with for space groups and for plane groups.
Examples are:
subgroups of index 2 for , No. 13, and , No. 196;
subgroup of index 2 for , No. 14, and , No. 143; ;
subgroups of index 2 for , No. 6, and , No. 2;
subgroups of index 2 for , No. 47.
Lemma 1.2.8.2.5. The number of isomorphic subgroups of each space group is infinite and this applies even to the number of maximal isomorphic subgroups.
Nevertheless, their listing is possible in the form of infinite series. The series are specified by parameters.
Lemma 1.2.8.2.6. For each space group, each maximal isomorphic subgroup can be listed as a member of one of at most four series of maximal isomorphic subgroups. Each member is specified by a set of parameters.
The series of maximal isomorphic subgroups are discussed in Section 2.1.5 .
References
International Tables for Crystallography (2005). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Heidelberg: Springer. (Abbreviated IT A.)