# Group algebra/matrix space homomorphism

I would like to know if it is possible to define a homomorphism (of algebras) between a group algebra over the rationals and a matrix space over the rationals.

I have

```
sage: S3 = SymmetricGroup(3)
sage: QG = GroupAlgebra(S3, QQ)
sage: QG.gens()
Finite family {(1,2,3): (1,2,3), (1,2): (1,2)}
sage: M_3 = MatrixSpace(QQ, 2)
sage: ma = M_3([[0,-1],[1,-1]])
sage: mb = M_3([[1,-1],[0,-1]])
```

Now I would like to map each generator of QG to a matrix. In this case I would like to map (1,2,3) to ma and (1,2) to mb and use that to define a homomorphism. But everything I tried didn't work.

Edit:

I've encountered another issue, though I can work around it. So this is only for information. The elements of `G.gens()`

don't coincide with the reduced words of length 1.

```
sage: S3 = SymmetricGroup(3)
sage: S3.gens()
[(1,2,3), (1,2)]
sage: {g: g.reduced_word() for g in S3}
{(): [],
(2,3): [2],
(1,2): [1],
(1,2,3): [2, 1],
(1,3,2): [1, 2],
(1,3): [1, 2, 1]}
```

Looks like a bug... I modified ma = matrix(...) to ma = M_3(...) and similarly for mb.

It even seems impossible to get a group morphism from S3 to MatrixGroup([ma,mb])... which looks like another bug!

yes, I used M_3 instead of matrix, I posted the wrong code here, sorry for that

about your edit: I would not characterize it as a bug. This is well documented, the reduced_word method returns a word over the simple reflections (1,2), (2,3), (3,4), ...