CARAT Introduction / Examples / Do two given groups have Z-equivalent copies which lie in a finite unimodular group?
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 9 - Do two given groups have Z-equivalent copies which lie in a finite unimodular group?
Keywords: CARAT, crystallography, groups, group, bravaisgroup, bravaisgroups, bravais groups, bravais group, mathematics, algorithms, algorithm, examples, example, Bravais_grp, Bravais_inclusions, Is_finite, Z-equivalent, finite, Z equivalent, arithmetically equivalent

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### Example 9

Do the groups G and H given below have Z-equivalent copies which lie in a finite unimodular group?

G is generated by

    6
    0 0 0 0 0 -1
    0 0 0 0 1 -1
    1 0 0 0 0  0
    0 1 0 0 0  0
    0 0 1 0 0  0
    0 0 0 1 0  0

and H is generated by

    6
    0 -1 0 0 0  0
    1  0 0 0 0  0
    0  0 0 0 0 -1
    0  0 0 0 1  0
    0  0 1 0 0  0
    0  0 0 1 0  0

### Used Programs

Bravais_grp, Bravais_inclusions (and Is_finite)

### Solution

1. Note first that the answer is positive if and only if G and H are contained in Z-equivalent Bravais groups. Therefore write each of the generating sets into files called 'g' and 'h' and make sure that you add "#g1" as top line to have the files in bravais_TYP format.
2. Call

                Bravais_grp g > gb
                Bravais_grp h > hb

   to write the Bravais groups of G and H into files 'gb' and 'hb'. (Note, we have assumed already that G and H are finite. By calling

                Is_finite g
                Is_finite h

   this could have been checked beforehand.)
3. Call

                Bravais_inclusions -S gb
                Bravais_inclusions -S hb

   to get lists of names for the Z-classes of all Bravais groups containing G resp. H. We get

                Bravais groups which contain a Z-equivalent subgroup
                Symbol: 6-2'  homogeneously d.: 1 zclass: 1
                Symbol: 6-2  homogeneously d.: 1 zclass: 1
                Symbol: 6-2  homogeneously d.: 2 zclass: 1
                Symbol: 6-2  homogeneously d.: 3 zclass: 1

   and

                Bravais groups which contain a Z-equivalent subgroup
                Symbol: 4-1';2-1  homogeneously d.: 1 zclass: 1
                Symbol: 4-1;2-1  homogeneously d.: 1 zclass: 1
                Symbol: 4-1;2-1  homogeneously d.: 2 zclass: 1
                Symbol: 6-1  homogeneously d.: 1 zclass: 1

   There are no common names. Hence there is no finite subgroup of GL_6(Z) containig G and a GL6(Z)-conjugate of H.

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last change: 11.09.2000