Constructing maps between cohomology groups in magma

Question

How does one construct maps between cohomology groups in $\texttt{magma}$? Here is a (randomly generated) example:

G:=SL(2,8);
V:=GModule(G);
M:=SymmetricPower(V,10);
S,phi:=Socle(M);

CM:=CohomologyModule(G,M);
CS:=CohomologyModule(G,S);
H1M:=CohomologyGroup(CM,1);
H1S:=CohomologyGroup(CS,1);

H1M;
Full Vector space of degree 1 over GF(2^3)
H1S;
Full Vector space of degree 1 over GF(2^3)

Both cohomology groups have dimension $1$ and $\phi: S \rightarrow M$ induces a map $\phi_*$ on cohomology. $\texttt{H1S}$ has a generator $\texttt{H1S.1}$; how do I compute the image of this class under $\phi_*$ using $\texttt{magma}$? The "obvious" guess

phi(H1S.1);

(not surprisingly for magma) does not work.

Answer 1

I think you have to define the induced map explicitly using cocycles.

> G:=SL(2,8);
> V:=GModule(G);
> M:=SymmetricPower(V,10);
> S,phi:=Socle(M);
> 
> CM:=CohomologyModule(G,M);
> CS:=CohomologyModule(G,S);
> H1M:=CohomologyGroup(CM,1);
> H1S:=CohomologyGroup(CS,1);
> 
> ImCocycle := func< occ | func< s | phi(occ(s)) > >;
> 
> phistar := hom< H1S->H1M |
>   x :-> IdentifyOneCocycle(CM,ImCocycle(OneCocycle(CS,x))) >;
> phistar(H1S.1);
(    1)

Perhaps this functionality should be defined within Magma. There is an example in the manual showing you how to define the restriction map from a group to subgroup,which is similar, but amore straightforward.