[GAP Forum] Direct Product

[Alexander Konovalov]

Dear Ian,

I was trying to reproduce your computations, and, if I was following
your way correctly, then

Image(Embedding(GDirect,1),b);

will not work because b belongs to the dihedral group of order 12
which is the 2nd factor of the direct product, not the 1st one.
Thus, to embed b into the direct product, you need to use

gap> Image(Embedding(GDirect,2),b);

See my working example below:

gap> GCyclic := CyclicGroup( 2 ) ;;
gap> GDihedral := DihedralGroup( 12 ) ;;
gap> GDirect := DirectProduct(GCyclic,GDihedral);;
gap> h := GeneratorsOfGroup( GCyclic   )[1] ;
f1
gap> a := GeneratorsOfGroup( GDihedral )[1] ;
f1
gap> b := GeneratorsOfGroup( GDihedral )[2] ;
f2
gap> Image(Embedding(GDirect,1),h);
f1
gap> Image(Embedding(GDirect,2),a);
f2
gap> Image(Embedding(GDirect,2),b);
f3

Hope it helps, and please do not hesitate to ask further questions.

Best wishes,
Alexander


On 16 Jan 2007, at 16:10, Ian McLoughlin wrote:

> Hi,
>
> I'm having a problem with creating a listing of the direct product  
> of two
> groups. They are...
>
> 1) The cyclic group(C2) of order 2:            < h | h^2 = 1 > and
> 2) The dihedral group(D12) of order 12:    < a , b | a^2 = 1 , b^6  
> = 1 , ab
> = ba^(-1) >.
>
> What I need is a listing of the elements of C2 x D12(in a list) in  
> the order
> that I want....
>
> [ 1 , b , b^2 , ... , b^5 , a , ab , ab^2 , ... , ab^5 , h , hb ,  
> hb^2 , ...
> , hb^5 , hab , hab^2 , hab^5 ].
>
> When I try to use DirectProduct and the Image(Embedding(GDirect,1),b);
> technique in other posts to this forum about direct products, I get an
> error. What i tried was....
>
>
> h := GeneratorsOfGroup( GCyclic    )[1] ;
> a := GeneratorsOfGroup( GDihedral )[1] ;
> b := GeneratorsOfGroup( GDihedral )[2] ;
>
> with
>
> GCyclic := CyclicGroup( 2 ) ;
> GDihedral := DihedralGroup( 12 ) ;
>
> .
>
> Thanks,
> Ian