[GAP Forum] Projection for Semidirect Product
Alexander Hulpke
hulpke at math.colostate.edu
Thu Oct 10 21:05:59 BST 2013
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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