[GAP Forum] Re: S4 Symmerty

[Alexander Hulpke]

Dear GAP-forum,

A user wrote:

>
> I am not a GAP expert. I have a data file containing a collections of
> point sets, for instance,
> {{1,2,3,4},{0,2,3,4}}
> {{0,2,3,6},{0,1,3,6}}
> ...
> Here a permutation of 0,1,2,3 determines the action on the other  
> points
> too. I would like to look those collections of sets which are not  
> related by the
> permutation group S4. How can I do it using GAP?
>

I interpret your question that you have a group and want to get  
representatives of the orbits.

First, permutation groups in GAP do not act on 0 -- it is probably  
easiest to use an editor to replace 0 by a number that is not used,  
say 9.

The second thing you will have to do is to convert your objects into  
GAP-objects (in an editor):

objects:=[
[[1,2,3,4],[9,2,3,4]],
[[9,2,3,6],[9,1,3,6]],

and so on
];
Read this file into GAP.

Next we want to convert the objects into sets of sets. This means  
every inner and outer list must be sorted. Probably the easiest way  
to do this is:

objects:=List(objects,i->Set(List(i,Set)));

Now create the group (be careful that 0 again has to be translated.  
e.g. if you had S4 on the points 0,1,2,3, you would make with our 9  
translation:

G:=Group((9,1,2,3),(9,1));

Then you can compute orbits, picking one element each gives you  
representatives.

orb:=Orbits(G,objects,OnSetsSets);
List(orb,i->i[1]);

Best,

     Alexander Hulpke