TITLE: Is there a CAS that can solve a given system of equations in a finite group algebra $kG$?
QUESTION [6 upvotes]: Let $k$ be a finite field with char$(k)=p>0$. Let $G$ be a finite group.
Consider the group algebra $kG$.
I would like to solve a given system of equations in $kG$.
Question:

Is there a computer algebra system that can solve a given system of equations in $kG$?

Any help is appreciated.
Thank you very much.

REPLY [2 votes]: A partial answer in a very special case, but still possibly useful to record.
The search for idempotents in a group algebra can be simplified by using GAP's MeatAxe: https://www.gap-system.org/Manuals/doc/ref/chap69.html
For example, if $G$ is the dihedral group with eight elements, and $F$ the field with eight elements, the following code verifies that $kG$ is indecomposable as a $G$-module, which means that the only idempotents are zero and one.
G := DihedralGroup(8)
k := FiniteField(8);
cayley := Action(G, AsList(G), OnRight);
reg := PermutationGModule(cayley, k);
MTX.IsIndecomposable(reg);

More generally, if $M$ is a MeatAxe module, you can compute a direct sum decomposition using
MTX.Indecomposition(M)

Of course, this doesn't solve for all the idempotents---it only returns a few.  You can also use
MTX.HomogeneousComponents(M)

to break the search into smaller algebras before using a more brute-force method.