# Re: [GAP Forum] working with GroupRings

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Dear Thomas and forum.```
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This is indeed what I'm trying to do.
Thanks for the code sample!

The Field() construction seems to behave
much nicer than AlgebraicExtension.

I'm going to push my luck a bit and ask
if I can adjoin 2^(1/3) since I want to work
in the splitting field of x^3-2.

As an aside, I found that back in 2001 or so
someone was trying to develop a Hopf algebra
package for GAP, but I couldn't find any further
mention of it. (I am looking at sub-algebras of
Hopf algebras.)

-T

On Thu, 26 Oct 2017, Thomas Breuer wrote:

> Dear Forum,
>
> concerning the question asked by Tim Kohl,
> about dealing with certain subspaces of group algebras,
> I am not sure whether I understand what the goal is.
>
> Perhaps we try a different approach, using the implementation
> of cyclotomic fields instead of the algebraic extensions.
> (Of course this is neither general enough nor practically desirable.)
>
> Consider the following GAP session.
>
>     gap> r2:= Sqrt( 2 );;                 # a square root of 2
>     gap> K:= Field( Rationals, [ r2 ] );; # the field extension
>     gap> S3:= SymmetricGroup( 3 );;
>     gap> KS3:= GroupRing( K, S3 );;
>     gap> emb:= Embedding( S3, KS3 );;
>     gap> gens:= [ (1,2)^emb + (1,2,3)^emb, (1,3,2)^emb + (1,3)^emb ];
>     [ (1)*(1,2)+(1)*(1,2,3), (1)*(1,3,2)+(1)*(1,3) ]
>
> Now one wants to create a Q-space that is generated by some elements in KS3.
> For that, we can either view the group algebra as a Q-space, ...
>
>     gap> Q_KS3:= AsAlgebra( Rationals, KS3 );;
>     gap> Dimension( Q_KS3 );
>     12
>     gap> Dimension( KS3 );
>     6
>     gap> H1:= Subspace( Q_KS3, gens );
>     <vector space over Rationals, with 2 generators>
>
> ... or we form the vector space independent of the group algebra,
> just by prescribing the base field and generators.
> (The two variants are of course equal as sets.)
>
>     gap> H2:= VectorSpace( Rationals, gens );
>     <vector space over Rationals, with 2 generators>
>     gap> H1 = H2;
>     true
>
> In both cases, forming products of elements in the subspaces works.
>
>     gap> prod:= gens[1] * gens[2];
>     (1)*()+(1)*(2,3)+(1)*(1,2)+(1)*(1,2,3)
>     gap> prod in H;
>     false
>
> Is this roughly the setup of interest?
> If yes then the analogous construction using general algebraic extensions
> would require to deal with algebras/spaces over subfields of the extension.
> Is that available?
>
> All the best,
> Thomas
>
>
> On Thu, Oct 26, 2017 at 11:15:16AM -0400, tk...@math.bu.edu wrote:
> >
> > Dear forum,
> >
> > This has gotten me part way to what I'm looking for.
> >
> > (Many thanks Frank.)
> >
> > But I'm running into a different problem now.
> >
> > Basically, if one has, for example,
> >
> > a:=Indeterminate(Rationals,"r");
> > K:=AlgebraicExtension(Rationals,r^2-2)
> > S3:=SymmetricGroup(3);
> > KS3:=GroupRing(K, S3);
> > emb:=Embedding(S3,KS3);
> >
> > then I would like to be able to view KS3 as
> > a module over Q, so that I can do something
> > like this
> >
> > H:=Subspace(KD3,[(1,2)^emb+(1,2,3)^emb, (1,3,2)^emb+(1,3)^emb ]);
> >
> > so that H is the Q-span of { (1,2)+(1,2,3) , (1,3,2)+(1,3) }
> > with the ultimate goal of being able to multiply elements
> > of H and represent them with respect to this basis.
> >
> > [I'm basically looking at Q-subalgebras of KG.]
> >
> > Also, as an aside, the GaloisGroup() function seems
> > not to be working. It gives the "no method found" error
> > if I try to do GaloisGroup(K).  [I'm using 4r8.]
> >
> > Thanks.
> >
> >     -T
>
>
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--
Dr. Timothy Kohl | Desktop Services Specialist, Sr.
Boston University Information Services & Technology | IT Help Center
tk...@bu.edu | 617.353.8203 | bu.edu/tech

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