CARAT Introduction / Examples / Determine the maximal finite subgroups of GL_5(Z)
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 3 - Determine the maximal finite subgroups of GL<sub>5</sub>(Z)
Keywords: CARAT, crystallography, groups, group, mathematics, algorithms, tables, algorithm, table, examples, example, Bravais_catalog, Datei, Bravais_inclusions, subgroups, subgroup, GL

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### Example 3

Determine the maximal finite subgroups of GL5(Z)

### Used Programs

Bravais_catalog/Datei, Bravais_inclusions

### Solution

1. Call

                Bravais_catalog

   Input family symbol 1,1,1,1,1, calculate the groups, choose file name 11111 and print all groups. The Bravais group <-I5> is written to file '11111'. Note, each Bravais group of degree 5 contains this group.
2. Call

                Bravais_inclusions 11111 -S > all

   Now the file 'all' contains a list of the names of all Bravais groups of degree 5, more precisely representatives of the Z-classes.
   (Note:

                grep Symbol all | wc

   would tell us that there are 189 Z-classes of Bravais groups of degree 5.)
3. Call

                Bravais_catalog

   Input family symbol 5-1, calculate the groups, choose file name 51 and print all groups. All Bravais groups in family 5-1 are now listed in file '51'.
   There are three Bravais groups in file '51' now. By irreducibility all three Bravais groups are maximal finite. Later we want to omit their proper Bravais subgroups from the file 'all'. To prepare this edit the file '51' and split it up into three files '51a', '51b' and '51c' containing one Bravais group each.
4. Call

                Bravais_catalog

   Input family symbol 5-2 calculate the groups, choose file name 52 and print all groups. All Bravais groups in family 5-1 are now listed in file '52'.
   There are four Bravais groups in file 52 now. By irreducibility all four Bravais groups are maximal finite. Later we want to omit their proper Bravais subgroups from the file 'all'. To prepare this edit the file '52' and split it up into four files '52a', '52b', '52c' and '52d' containing one Bravais group each.
5. Call

                Bravais_inclusions 51a > notmax
                Bravais_inclusions 51b >> notmax
                Bravais_inclusions 51c >> notmax
                Bravais_inclusions 52a >> notmax
                Bravais_inclusions 52b >> notmax
                Bravais_inclusions 52c >> notmax
                Bravais_inclusions 52d >> notmax

   to write the names of the Bravais subgroups of the five maximal finite subgroups known so far on file 'notmax'.
6. Call

                grep Symbol notmax > compare
                sort -u compare > notmax

   Now the file 'notmax' contains the names of the seven maximal groups and of their proper Bravais subgroups, each one listed just once. By editing the file 'notmax', one writes the lines corresponding to maximal groups on a new file 'MAX. (These are the last seven lines.)
7. Call

                sort all > allsort
                diff allsort notmax | grep Symbol > all

   Now the file 'all' contains a complete list of Bravais groups, which are not contained in one of the groups of the file 'MAX'.
8. The last lines of the file 'all', more precisely the ones involving a symbol of the form 4-x;1 are the following:

                < Symbol: 4-1;1  homogeneously d.: 2 zclass: 1
                < Symbol: 4-2';1  homogeneously d.: 1 zclass: 1
                < Symbol: 4-2;1  homogeneously d.: 1 zclass: 1
                < Symbol: 4-2;1  homogeneously d.: 2 zclass: 1
                < Symbol: 4-3';1  homogeneously d.: 1 zclass: 1
                < Symbol: 4-3;1  homogeneously d.: 1 zclass: 1
                < Symbol: 4-3;1  homogeneously d.: 2 zclass: 1

   By using

                Bravais_inclusions -S

   one can rule out the two groups involving a ' as maximal finite groups. For the remaining five groups it is clear now that they are maximal finite. So by repeating the above computations with these five groups leads us to a new file 'MAX' containig 7+5 groups and a new file 'all' containing all Bravais groups not contained in any of the groups in 'MAX' (up to Z-equivalence). It turns out that the process terminates after the next step with 'MAX' looking like

                5-1a       5-1c       5-2b       5-2d
                5-1b       5-2a       5-2c
                4-112   4-211   4-212   4-311   4-312
                32-11   32-13   32-21   32-22   32-23

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last change: 11.09.2000