The GAP package NormalizersOfPrimitiveGroups

NormalizersOfPrimitiveGroups is a GAP package to compute normalizers of primitive groups.

Note: Currently, the package can only be loaded using the master branch of GAP.

You can use it as follows:

1. Download NormalizersOfPrimitiveGroups and extract it into a GAP pkg directory.

2. Start GAP and load the NormalizersOfPrimitiveGroups package:

   LoadPackage("NormalizersOfPrimitiveGroups");

3. You can use the function NormalizerInSymmetricGroupOfPrimitiveGroup to compute the normalizer of a primitive group inside the symmetric group on the same number of points.

4. You can use the function WeakCanonizerOfPrimitiveGroup to compute a permutation which conjugates a given primitive group into weak canonical form and the corresponding socle-component.

Note: NormalizerInSymmetricGroupOfPrimitiveGroup and WeakCanonizerOfPrimitiveGroup are currently only implemented for groups of type PA.

Example

The following is an example session using this package to compute for a primitive group of type PA:

• the weak canonical form
• the normalizer in the ambient symmetric group

gap> LoadPackage("NormalizersOfPrimitiveGroups");;
# Construct a primitive group of type PA via the function
# WreathProductProductAction
gap> T := AlternatingGroup(5);;
gap> m := NrMovedPoints(T);;
gap> d := 2;;
gap> top := Group((1,2));
gap> WP := WreathProductProductAction(T, top);;
# Conjugate this group with a random permutation
gap> random := Random(SymmetricGroup(m ^ d));;
gap> G := WP ^ random;;
# Notice how WeakCanonizerOfPrimitiveGroup transforms the conjugate back into a
# product action wreath product.
# We omit the explicit permutation and group
gap> r := WeakCanonizerOfPrimitiveGroup(G);
rec(
    conjugatorToWeakCanonicalForm := ...,
    socleComponent := ...
 )
gap> GeneratorsOfGroup(G ^ r.conjugatorToWeakCanonicalForm);
[ (1,5,2,4,3)(6,10,7,9,8)(11,15,12,14,13)(16,20,17,19,18)(21,25,22,24,23),
  (1,6,11,16,21)(2,7,12,17,22)(3,8,13,18,23)(4,9,14,19,24)(5,10,15,20,25),
  (1,5,2)(6,10,7)(11,15,12)(16,20,17)(21,25,22),
  (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15),
  (2,11)(3,21)(4,16)(5,6)(7,15)(8,25)(9,20)(13,22)(14,17)(18,24) ]
# Compute the normalizer with our algorithm and compare it to the built-in
# method
gap> N := NormalizerInSymmetricGroupOfPrimitiveGroup(G);;
gap> N = Normalizer(SymmetricGroup(25), G);
true

Generating more examples

If you want more examples of PA type try the following code. This does not generate all PA type groups with a given socle though, since subgroups of those groups can still be primitive.

gap> m := 5;;
gap> d := 3;;
gap> T := OnePrimitiveGroup(NrMovedPoints, m, ONanScottType, "2");;
gap> top := OneTransitiveGroup(NrMovedPoints, d);;
gap> WP := WreathProductProductAction(T, top);;

Documentation

Run the makedoc.g script to build the documentation.

Note: It is currently not possible to build the documentation due to incompatibilities between AutoDoc and the current GAP master branch.

Tests

For the package tests run any of the files tst/teststandard.g tst/teststandard-with-asserts.g tst/testall.g with the current master branch of GAP.

Contact

Please submit bug reports, suggestions for improvements and patches via the issue tracker or via email to Sergio Siccha.

License

NormalizersOfPrimitiveGroups is free software you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the file LICENSE distributed as part of this package or see the FSF's own site.