[GAP Forum] Projection for Semidirect Product

[Alexander Hulpke]

On Oct 10, 2013, at 10/10/13 1:44, Sopsku <rrburns at cox.net> wrote:

> 
> Sorry for being so dense, but I do not fully understand the command and it

Indeed, it should have been (swapped indices 1,2)

PreImagesRepresentative(Embedding(s,2), r/Image(Embedding(s,1),Image(Projection(s),r));

> fails so I am not quite sure what Prof. Hulpke intended. I did try the
> following:

Assuming your setup is:

gap> g:=Group((1,2,3,4,5));
Group([ (1,2,3,4,5) ])
gap> a:=AutomorphismGroup(g);
<group with 1 generators>
gap> s:=SemidirectProduct(a,g);
<pc group with 3 generators>
gap> p:=Projection(s);
[ f1, f2, f3 ] -> [ [ (1,2,3,4,5) ] -> [ (1,3,5,2,4) ], 
  [ (1,2,3,4,5) ] -> [ (1,5,4,3,2) ], IdentityMapping( Group([ (1,2,3,4,
   5) ]) ) ]
gap> e1:=Embedding(s,1);
CompositionMapping( [ f1, f2 ] -> [ f1, f2 ], CompositionMapping( Pcgs(
[ (1,2,4,3), (1,4)(2,3) ]) -> [ f1, f2 ], <action isomorphism> ) )
gap> e2:=Embedding(s,2);
[ (1,2,3,4,5) ] -> [ f3 ]


> npart1:=function(elm)
>   return Image(e1,Image(p,elm))^p;
> end;
> 
> npart2:=function(elm)
>   return PreImagesRepresentative(e2,elm/Image(e1,Image(p,elm)));
> end;

> 
> PrintArray(List(Elements(S),e->[npart1(e),npart2(e)]));
> 
> Which is more or less what I think I am looking for. Is this kind of what
> Prof. Hulpke intended.

Yes, though your function npart1 does nothing else but just computing  Image(p,elm)

Best,

Alexander Hulpke

> 
> Again thank you for any help or comments.
>    Ron
> 
> 
> 
> 
> 
> 
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