Graphs for translationengleiche subgroups

Wondratschek, H.; Aroyo, M. I.

doi:10.1107/97809553602060000542

International
Tables for
Crystallography
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

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International Tables for Crystallography (2006). Vol. A1. ch. 2.1, pp. 54-55 | 1 | 2 |

Section 2.1.7.2. Graphs for translationengleiche subgroups

Hans Wondratschek^a ^* and Mois I. Aroyo^b

^a Institut für Kristallographie, Universität, D-76128 Karlsruhe, Germany, and ^bDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain
Correspondence e-mail: wondra@physik.uni-karlsruhe.de

2.1.7.2. Graphs for translationengleiche subgroups

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Let $[{\cal G}]$ be a space group and $[{\cal T}]$ ( $[{\cal G}]$ ) the normal subgroup of all its translations. Owing to the isomorphism between the factor group $[{\cal G}/{\cal T}({\cal G})]$ and the point group $[{\cal P}_{{\cal G}}]$ , see Section 1.2.5.4 , according to the first isomorphism theorem, Ledermann (1976), t-subgroup graphs are the same (up to the symbols) as the corresponding graphs between point groups. However, in this volume, the graphs are not complete but are contracted by displaying each space-group type at most once. This contraction may cause the graphs to look different from the point-group graphs and also different for different space groups of the same point group, cf. Example 2.1.7.2.1.

One can indicate the connections between a space group $[{\cal G}]$ and its maximal subgroups in different ways. In the contracted t-subgroup graphs one line is drawn for each conjugacy class of maximal subgroups of $[{\cal G}]$ . Thus, a line represents the connection to an individual subgroup only if this is a normal maximal subgroup of $[{\cal G}]$ , otherwise it represents the connection to more than one subgroup. The conjugacy relations are not necessarily transferable to non-maximal subgroups, cf. Example 2.1.7.2.2. On the other hand, multiple lines are possible, see the examples. Although it is not in general possible to reconstruct the complete graph from the contracted one, the content of information of such a graph is higher than that of a graph which is drawn with simple lines only.

The graph for the space group at its top also contains the contracted graphs for all subgroups which occur in it, see the remark below Example 2.1.7.2.2.

Owing to lack of space for the large graphs, in all graphs of t-subgroups the group [P1] , No. 1, and its connections have been omitted. Therefore, to obtain the full graph one has to supplement the graphs by [P1] at the bottom and to connect [P1] by one line to each of the symbols that have no connection downwards.

Within the same graph, symbols on the same level indicate subgroups of the same index relative to the group at the top. The distance between the levels indicates the size of the index. For a more detailed discussion, see Example 2.1.7.2.2. For the sequence and the numbers of the graphs, see the paragraph below Example 2.1.7.2.2.

Example 2.1.7.2.1

Compare the t-subgroup graphs in Figs. 2.4.4.2 , 2.4.4.3 and 2.4.4.8 of [Pnna] , No. 52, [Pmna] , No. 53, and [Cmce] , No. 64, respectively. The complete (uncontracted) graphs would have the shape of the graph of the point group [mmm] with [mmm] at the top (first level), seven point groups⁵ ( [222] , [mm2] , [m2m] , [2mm] , [112/m] , [12/m1] and [2/m11] ) in the second level, seven point groups ( [112] , [121] , [211] , [11m] , [1m1] , [m11] and $[\overline{1}]$ ) in the third level and the point group [1] at the bottom (fourth level). The group [mmm] is connected to each of the seven subgroups at the second level by one line. Each of the groups of the second level is connected with three groups of the third level by one line. All seven groups of the third level are connected by one line each with the point group 1 at the bottom.

The contracted graph of the point group [mmm] would have [mmm] at the top, three point-group types ( [222] , [mm2] and [2/m] ) at the second level and three point-group types ( [2] , m and $[\overline{1}]$ ) at the third level. The point group 1 at the bottom would not be displayed (no fourth level). Single lines would connect [mmm] with 222, [mm2] with 2, [2/m] with 2, [2/m] with m and [2/m] with $[\overline{1}]$ ; a double line would connect [mm2] with m; triple lines would connect [mmm] with [mm2] , [mmm] with [2/m] and 222 with 2.

The number of fields in a contracted t-subgroup graph is between the numbers of fields in the full and in the contracted point-group graphs. The graph in Fig. 2.4.4.2 of [Pnna] , No. 52, has six space-group types at the second level and four space-group types at the third level. For the graph in Fig. 2.4.4.3 of [Pmna] , No. 53, these numbers are seven and five and for the graph in Fig. 2.4.4.8 of [Cmce] , No. 64 (formerly [Cmca] ), the numbers are seven and six. However, in all these graphs the number of connections is always seven from top to the second level and three from each field of the second level downwards to the ground level, independent of the amount of contraction and of the local multiplicity of lines.

Example 2.1.7.2.2

Compare the t-subgroup graphs shown in Fig. 2.4.1.1 for $[Pm\overline{3}m]$ , No. 221, and Fig. 2.4.1.5 , $[Fm\overline{3}m]$ , No. 225. These graphs are contracted from the point-group graph $[m\overline{3}m]$ . There are altogether nine levels (without the lowest level of [P1] ). The indices relative to the top space groups $[Pm\overline{3}m]$ and $[Fm\overline{3}m]$ are 1, 2, 3, 4, 6, 8, 12, 16 and 24, corresponding to the point-group orders 48, 24, 16, 12, 8, 6, 4, 3 and 2, respectively. The height of the levels in the graphs reflects the index; the distances between the levels are slightly distorted in order to adapt to the density of the lines. From the top space-group symbol there are five lines to the symbols of maximal subgroups: The three symbols at the level of index 2 are those of cubic normal subgroups, the one (tetragonal) symbol at the level of index 3 represents a conjugacy class of three, the symbol $[R\overline{3}m]$ , No. 166, at the level of index 4 represents a conjugacy class of four subgroups.

The graphs differ in the levels of the indices 12 and 24 (orthorhombic, monoclinic and triclinic subgroups) by the number of symbols (nine and seven for index 12, five and three for index 24). The number of lines between neighbouring connected levels depends only on the number and kind of symbols in the upper level. This property makes such graphs particularly useful.

However, for non-maximal subgroups the conjugacy relations may not hold. For example, in Fig. 2.4.1.1 , the space group [P222] has three normal maximal subgroups of type [P2] and is thus connected to its symbol by a triple line, although these subgroups are conjugate subgroups of the non-minimal supergroup $[Pm\overline{3}m]$ .

The t-subgroup graphs in Figs. 2.4.1.1 and 2.4.1.5 contain the t-subgroups of $[Pm\overline{3}m]$ (221) and $[Fm\overline{3}m]$ (225) and their relations. In addition, the t-subgroup graph of $[Pm\overline{3}m]$ includes the t-subgroup graphs of [P432] , $[P\overline{4}3m]$ , $[Pm\overline{3}]$ , [P23] , [P4/mmm] , $[P\overline{4}2m]$ , $[P\overline{4}m2]$ , [P4mm] , $[R\overline{3}m]$ , [R3m] etc., that of $[Fm\overline{3}m]$ includes those of [F432] , $[F\overline{4}3m]$ , $[Fm\overline{3}]$ , [ I4/mmm] , also $[R\overline{3}m]$ etc. Thus, many other graphs can be extracted from the two basic graphs. The same holds for the other graphs displayed in Figs. 2.4.1.2 to 2.4.4.8 : each of them includes the contracted graphs of all its subgroups. For this reason one does not need 229 or 218 different graphs to cover all t-subgroup graphs of the 229 space-group types but only 37 ( [P1] can be excluded as trivial).

The preceding Example 2.1.7.2.2 suggests that one should choose the graphs in such a way that their number can be kept small. It is natural to display the `big' graphs first and later those smaller graphs that are still missing. This procedure is behind the sequence of the t-subgroup graphs in this volume.

(1) The ten graphs of $[Pm\overline{3}m]$ , No. 221, to $[Ia\overline{3}d]$ , No. 230, form the first set of graphs in Figs. 2.4.1.1 to 2.4.1.10 .
(2) There are a few cubic space groups left which do not appear in the first set. They are covered by the graphs of (213), (212) and $[Pa\overline{3}]$ (205). These graphs have large parts in common so that they can be united in Fig. 2.4.1.11 .
(3) No cubic space group is left now, but only eight tetragonal space groups of crystal class have appeared up to now. Among them are all graphs for space groups with an I lattice which are contained in Figs. 2.4.1.5 to 2.4.1.8 of the F-centred cubic space groups. The next 12 graphs, Figs. 2.4.2.1 to 2.4.2.12 , are those for the space groups of the crystal class with lattice symbol P and different third and fourth constituents of the HM symbol. They start with , No. 124, and end with , No. 138.
(4) Two (enantiomorphic) tetragonal space-group types are left which are compiled in Fig. 2.4.2.13 .
(5) The next set is formed by the four graphs in Figs. 2.4.3.1 to 2.4.3.4 of the hexagonal space groups , No. 191, to , No. 194. The hexagonal and trigonal enantiomorphic space groups do not appear in these graphs. They are combined in Fig. 2.4.3.5 , the last one of hexagonal origin.
(6) Several orthorhombic space groups are still left. They are treated in the eight graphs in Figs. 2.4.4.1 to 2.4.4.8 , from , No. 51, to , No. 64 (formerly ).
(7) For each space group, the contracted graph of all its t-subgroups is provided in at least one of these 37 graphs.

For the index of a maximal t-subgroup, Lemma 1.2.8.2.3 is repeated: the index of a maximal non-isomorphic subgroup $[{\cal H}]$ is always 2 for oblique, rectangular and square plane groups and for triclinic, monoclinic, orthorhombic and tetragonal space groups $[{\cal G}]$ . The index is 2 or 3 for hexagonal plane groups and for trigonal and hexagonal space groups $[{\cal G}]$ . The index is 2, 3 or 4 for cubic space groups $[{\cal G}]$ .

References

Ledermann, W. (1976). Introduction to group theory. London: Longman. (German: Einführung in die Gruppentheorie, Braunschweig: Vieweg, 1977.)Google Scholar

International Tables for Crystallography (2006). Vol. A1. ch. 2.1, pp. 54-55