Graph and Chains of Maximal Subgroups
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Each group-subgroup pair of space groups can be represented as a chain of maximal
subgroups that relate the two groups in the pair. The program SUBGROUPGRAPH
permits to obtain:
- All of the possible chains of maximal subgroups that relate two space
groups with unspecified index -
Graph of maximal subgroups.
- The chains of maximal subgroups that relate two space groups with
specified index -
Chains of maximal subgroups
- The classification of the different subgroups of the same type for
a given space group with specified index -
Classes of subgroups
To obtain the graph of maximal subgroups relating two space groups you should
give the numbers of these two group, as given in the
International Tables for Crystallography, Vol. A or you can select them.
No value for the index should be given. To obtain the graph, click on
[Construct the graph].
Let G be the group and H the subgroup. The list of maximal subgroups that
relates the groups G and H is represented in a table.
The first row contains the group G and its maximal subgroups, given by their
numbers in the International Tables for Crystallography, Vol. A, and the
corresponding indices, given in brackets. The last row contains the same
information about the subgroup H. The rest of the rows in the table
contain the group number and symbol and a list with the maximal subgroups
and their indices
for all of the maximal subgroups that appear between G and H. If a chain
relating G and H is represented as
G > Z_{1} > .. > Z_{i} > ... Z_{n} > H
then each row of the table starts with the number and the Hermann-Mauguin
symbol of a group Z_{k} and contains the list with the groups
Z_{j} < Z_{k} that can appear in the graph relating G and H.
The results are graphically represented using the button [Draw the graph].
All of the programs need as an input the number of one or two space groups
as given in International Tables for Crystallography, Vol. A. If you do
not know these numbers, you can select them from the
Table of Space Group Symbols.
To obtain the chains of maximal subgroups that relate a group G with its
subgroup H with specified index, you should give (or select from the
table with group symbols) the group and the subgroup numbers as given in the
International Tables for Crystallography, Vol. A. Also, an index of the
subgroup H in G must be given.
To obtain the graph for the specified index,
click on [Construct the graph].
The resultant table contains all of the chains
G > Z_{1} > .. > Z_{i} > ... Z_{n} > H
that relate the group G with the subgroup H with the given index, represented
using the groups numbers and the Hermann-Mauguin symbols, and a link
transformation that shows all of the transformation matrices that
relate the basis of the group with that of the subgroup, obtained for the
current chain.
If you want to print only the table with the chains, follow the link
"Print this table".
In this case the graph is a part from the bigger graph that should be
obtained if the index is not specified.
You can obtain the graph using the button [Draw the graph].
Once you have obtained all of the chains that relate the group and the subgroup
with a specified index, you can go further and see how the different subgroups
of the same type as H are distributed into classes of conjugate groups. To do
that, click on [Classify (with a complete graph of all subgroups)].
The different classes of subgroups are given as tables, one table for each
class, which contain:
- the chain from which the subgroup has been obtained,
- the transformation matrix, that corresponds to that chain
- and a link to other chains (if some) that give the same subgroup.
To see the general positions of the subgroup with resect to the basis of the
group G, use the button in the column Transform with of the table.
All of the chains that will give the same group can be seen using the button in
the column Identical.
The graphical representation of the chains is a graph which
starts with the group G and ends with the subgroup H. The intermediate vertices
correspond to the groups Z_{i} that appear between G and H. A group is
connected with itself (loop edge) if it has isomorphic subgroups.
NOTE, that if the index is large then it is possible that the graph results
very complicated and difficult to use. If the graph is very big and can not
be seen with the browser you can use the PostSript form and see it with a
program for reading PostSript files.
Graph of Maximal Subgroups
In the graph of maximal subgroups there is one vertex for each of the groups
Z_{i} and for G and H. The index of H in G for each one of the possible
paths to relate them is obtained by multiplying the indices on each step in the
chain.
Chains of Maximal Subgroups
When the index of the subgroup in the group is specified, then the resultant graph contains only chains of maximal subgroups that correspond to the given
index.
Classes of Subgroups
The graph representing the classification if the different subgroups contains
not only the types of the subgroups but also all of the different subgroups of
the same type.
As a part of the label for the vertices corresponding to the subgroup H is given
the number of the class the current subgroup belongs to.
For each one of the chains there is a set of transformation matrices that can be
obtained following the chain.
If the chain is
G > Z_{1} > .. > Z_{i} > ... Z_{n} > H
and (P_{i}, p_{i}) is the transformation matrix that relates the
group Z_{i-1} with its maximal subgroup Z_{i}, then the matrix
that relates the basis of G with that of H for this chain is obtained using
(P,p) = (P_{1}, p_{1}) (P_{2}, p_{2})
...(P_{n+1}, p_{n+1}),
where (P_{n+1}, p_{n+1}) is the matrix corresponding to
Z_{n} > H.
The set of transformations contains all of the matrices that can be obtained
for a given chain.
If you have called the program SUBGROUPGRAPH
from other program (for example form WYCKSPLIT) than
for each one of the matrices there is a link to that program, so you can continue
using it with the data obtained from
SUBGROUPGRAPH.
It is possible that different chains of maximal subgroups give the same
subgroup. If you click on the button given in the column Identical of
the table with the subgroups in a given class, you can see the list with all of
the chains that will give the same subgroup. These chains are represented as a
table which contains the chain and the transformation matrix obtained following
this chain. Also, if you click on the button given in the column
Transform with you can obtain the general positions of the subgroup in
the basis of the supergroup, transformed with the current matrix.
More about the program