Let $G$ be a finite group and $k$ be a finite field of characteristic $p>0$ such that $p\mid |G|$.


Let $M$ be a $kG$-module which has an embedding $M\hookrightarrow kG^{reg}$ into the regular $kG$-module $kG^{reg}$.


Then $M$ corresponds to a right ideal of $kG$.


**Question:**



> 
> Is there a MAGMA command / procedure that has as input the $kG$-module $M$ and as output the corresponding right ideal $I$ in terms of giving me elements of the algebra $kG$ that generate $M$ as the right ideal $I$?
> 
> 
> 


I only could find how to get a basis of the module $M$ in the manual, but not how to transform this into the elements of the algebra $kG$.


More specifically, I am in the following concrete situation:


Let $P$ be a non-trivial $p$-subgroup of $G$. Let $T$ be the trivial group.


Consider the $kP$-module $M:= k \uparrow\_T^{P}\cong {kP}^{\text{reg}}$.


$M$ is a permutation module. Hence, $k\_P$ is a submodule of $M$ (namely, $\{\lambda\cdot (1,1,1,\dots , 1)^t | \lambda\in k\} \cong k$, if $M$ is given as a representation via permutation matrices).


Since $kG$ is free as $kP$-module, the induction functor is exact.


Thus, via induction, we get the $kG$-module $N:=k\uparrow\_P^G$ as a submodule of $M\uparrow\_P^G\cong {kG}^{\text{reg}}$.


I would like to get $N$ (and all of its direct summands) as a right ideal of $kG$ and couldn't find out how to do this in MAGMA.


Cross-reference: <https://math.stackexchange.com/questions/3848128/magma-question-concerning-the-transformation-of-a-kg-module-m-into-a-right-i>


I would be very grateful for any help.


Thanks in advance.



MAGMA-question concerning the transformation of a $kG$ -module $M$ into a right ideal of the group algebra
I am not sure if I understand completely what you are trying to do, but I get the impression that the heart of the problem is that you are given $kH$-module homomorphism $M \to N$, and you want to compute the induce homomorphism $M\_H^G \to N\_H^G$. I think the following code does that.



```
InducedHom := function(phi, G)
  //phi:M->N is a KH-module M->N homomorphism with H <= G
  //Compute induced homomorphism Induction(M,G) -> Induction(N,G)
  M := Domain(phi); N := Codomain(phi); K := BaseRing(M);
  H := Group(M);
  IM := Induction(M,G); IN := Induction(N,G);
  ind := Index(G,H);
  dm := Dimension(M); dn := Dimension(N);
  images := [];
  for i in [1..ind] do
    for j in [1..dm] do
      imseq := [K!0 : k in [1..(i-1)*dn]] cat Eltseq(phi(M.j)) cat
               [K!0 : k in [1..(ind-i)*dn]];
      Append(~images, IN!imseq);
    end for;
  end for;
  return hom<IM->IN | images>;
end function;

```

