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I have the following code written in GAP. In summary, In this code, I have a subgroup defined as SL(2,5). I applied the tensor product to each of these group elements with the identity and added a new element to the group, which is not in SL(2,5). Originally, I wanted to add a matrix that does not belong to SL(2,5), but GAP gave me an error. So, I had to write it over field 5, which forced everything into this characteristic and made it finite (Thanks for the warning @ahulpke).

After that, I checked if the new group is finite or not, and I also calculated the number of elements in this group. The thing is, I want to do the exact same thing with characteristic zero. However, I do not know how to do that. Originally, I wrote this code in Sagemath, but then I thought Sagemath might be more limited than GAP. That is why I converted my code from Sagemath to GAP. Now I want to make the characteristic 0 because my addition matrix will be an SU(4) matrix, and it should not have characteristic 5.

Id := IdentityMat;
Addition := [[1,0,0,0],[0,1,0,0],[1,0,0,1],[1,0,1,0]]*Z(5)^0;
new_matrices_ := [];
gens_ := GeneratorsOfGroup(SL(2,5)); # the subgroup you are interested in

for m in gens_ do
    tensor_mat_ := KroneckerProduct(m, Id(2)*Z(5)^0); # Id(2)
    Add(new_matrices_, tensor_mat_);
od;
Add(new_matrices_, Addition);

G := Group(new_matrices_);
Print(IsFinite(G));
Print(Size(G));
Print(StructureDescription(G));
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1 Answer 1

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I don't know whether this is what you want, but you can get a $2$-dimensional group G0 isomorphic to ${\rm SL}(2,5)$ in characteristic $0$ (with entries in the cyclotomic field of degree $5$) by:

G := SL(2,5);
I := IrreducibleRepresentations(G);
G0 := Image(I[2]);
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  • $\begingroup$ Thank you so much for the answer. When I update my code with your solution, I also rewrite identity as Id(2) instead of Id(2)*Z(5)^0 . But then I got the following error: Error, Variable: 'Id' must have an assigned value in tensor_mat_ := KroneckerProduct( m, Id( 2 ) ); at zero_characteristic.g:10 called from <function "unknown">( <arguments> ). Should I keep Identity matrix as it is? $\endgroup$
    – j.doe
    Commented Jul 21, 2023 at 0:31
  • $\begingroup$ the issue is resolved! $\endgroup$
    – j.doe
    Commented Jul 21, 2023 at 10:56

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