Klassengleiche
             or k subgroups of a space group

Wondratschek, H.

doi:10.1107/97809553602060000516

International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. A. ch. 8.3, p. 735

Section 8.3.3.2. Klassengleiche or k subgroups of a space group

H. Wondratschek^a ^*

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail: hans.wondratschek@physik.uni-karlsruhe.de

8.3.3.2. Klassengleiche or k subgroups of a space group

| top | pdf |

Every space group $[{\cal G}]$ has an infinite number of maximal k subgroups. For dimensions 1, 2 and 3, however, it can be shown that the number of maximal k subgroups is finite, if subgroups belonging to the same affine space-group type as $[{\cal G}]$ are excluded. The number of maximal subgroups of $[{\cal G}]$ belonging to the same affine space-group type as $[{\cal G}]$ is always infinite. These subgroups are called maximal isomorphic subgroups. In Part 13 isomorphic subgroups are treated in detail. In the space-group tables, only data on the isomorphic subgroups of lowest index are listed. The way in which the isomorphic and non-isomorphic k subgroups are listed in the space-group tables is described in Section 2.2.15 .

Remark: Enantiomorphic space groups have an infinite number of maximal isomorphic subgroups of the same type and an infinite number of maximal isomorphic subgroups of the enantiomorphic type.

Example

All k subgroups $[{\cal G}']$ of a given space group $[{\cal G} \equiv P3_{1}]$ , with basis vectors $[{\bf a}' = {\bf a},{\bf b}' = {\bf b},{\bf c}' = p{\bf c}, p]$ being any prime number except 3, are maximal isomorphic subgroups. They belong to space-group type $[P3_{1}]$ if [p = 3r + 1, r] any integer. They belong to the enantiomorphic space-group type $[P3_{2}]$ if [p = 3r + 2] .

Even though in the space-group tables some kinds of maximal subgroups are listed completely whereas others are listed only partly, it must be emphasized that in principle there is no difference in importance between t, non-isomorphic k and isomorphic k sub-groups. Roughly speaking, a group–subgroup relation is `strong' if the index [i] of the subgroup is low. All maximal t and maximal non-isomorphic k subgroups have indices less than four in $[E^{2}]$ and five in $[E^{3}]$ , index four already being rather exceptional. Maximal isomorphic k subgroups of arbitrarily high index exist for every space group.

References

International Tables for Crystallography (2006). Vol. A. ch. 8.3, p. 735