CARAT Introduction / Examples / Is the normalizer of a given group finite?

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

Create a file 'group' containing the above matrix and add
```
   #g1
```
as top line, to have a bravais_TYP file of the group G.
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.
Test, if 'norm' is finite: Call
```
   Is_finite norm
```
to find that the normalizer is infinite.

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.

last change: 11.09.2000