## Program Normalizer

### Usage

Normalizer 'file1' ['file2' ['file3']] [-p=prime] [-o]

file1: bravais_TYP containing the group G.

file2: (OPTIONAL) bravais_TYP containing the transposed of G. (cf.
Tr_bravais)

file3: (OPTIONAL) matrix_TYP of a G-perfect form. (cf. First_perfect)

### Description

Calculates a set of matrices which together with G generate the
normalizer N_{GLn(Z)}(G) of G in GL_{n}(Z).

NOTE: the output echoes any input information about G, except input about
generators of the normalizer.

NOTE: The dimension of the space of invariant forms is a measure for the
complexity of the algorithm. Up to degree 6 the only infeasible case are
<I_{6}> and <-I_{6}>. Here the generators of the normalizer can be taken
from Bravais_catalog.html/Datei
with family 1,1,1,1,1,1.
### Options

-b : The normalizer of the bravais group B(G) is calculated. With this
option the program is much faster. (The normalizer of G is a
subgroup of N_GL_n(Z) (B(G)). )
-p=prime: The determinants of the perfect forms are
calculated module prime. The default is 1949.
-o : The G-perfect forms are given as additional output.

### Remarks

See also for Bravais_catalog.html/Datei, First_perfect and Tr_bravais.
### Examples

- Find all space groups with a given point group and decide for
which superlattices each extension splits.
- Is the normalizer of a given group finite?