CARAT Introduction / Examples / Is the normalizer of a given group finite?
Author: Oliver Heidbüchel
Description: CARAT Introduction: Example 7 - Is the normalizer of a given group finite?
Keywords: CARAT, crystallography, group, mathematics, algorithms, tables, algorithm, table, examples, example, Is_finite, Normalizer, Symbol, finite, normalizer

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### Example 7

Does the group G generated by the following matrix have finite normalizer in GL6(Z)?

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

### Used Programs

Is_finite, Normalizer, Symbol

### Solution

1. Create a file 'group' containing the above matrix and add

          #g1

   as top line, to have a bravais_TYP file of the group G.
2. Call

         Normalizer group > norm

   to compute the normalizer of G and write G and its normalizer in bravais_TYP format on the file 'norm'. It turns out that the normalizer of G is generated by G and ten further matrices. Edit the file 'norm' to get a file in bravais_TYP containing the 11 generators of the normalizer; i. e. change the top line to

          #g11

   and delete the invariant forms.
3. Test, if 'norm' is finite: Call

          Is_finite norm

   to find that the normalizer is infinite.

#### Remark

Note, if one knows the families well, there is a quicker solution, because the finiteness of the normalizer is a family invariant. The call

       Symbol group

produces the answer

       4-1';2-1

which shows that the normalizer must be infinite.

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last change: 11.09.2000