# The normalized unit group using GAP.

I want the structure of The normalized unit group using GAP for the group algebra $$FD_{30}$$, where $$F$$ is a finite field with characteristic $$3$$ and $$D_{30}$$ is the dihedral group of order $$30.$$ I tried as follow.

true

gap> G:=DihedralGroup(30);;

gap> FG:=GroupRing(GF(3),G);

<algebra-with-one over GF(3), with 3 generators>

gap> IsGroupAlgebra(FG);

true

gap> V:=NormalizedUnitGroup(FG);

Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 2nd choice method found for NormalizedUnitGroup' on 1 arguments at /opt/homebrew/Cellar/gap/4.12.2/libexec/lib/methsel2.g:249 called from <function "HANDLE_METHOD_NOT_FOUND">( ) called from read-eval loop at stdin:5 type 'quit;' to quit to outer loop brk>

• What is the normalized unit group? A subgroup of the units? Commented Apr 2, 2023 at 1:11
• @ahulpke yes sir subgroup of unit group of augmentation $1$. Commented Apr 2, 2023 at 3:03
• Actually I want the structure of $1+J(F_{3^k}D_{30})$ where $J$ stand for the Jacobson radical. Commented Apr 2, 2023 at 3:06
• LAGUNA does not support such case - it's limited to modular group algebras of finite p-groups. Commented Apr 3, 2023 at 12:55

Try this? $$\color{red}{\text{I'm not sure whether the result is true}}$$

The ComputeNormalizedUnitGroup function takes a finite group G and a group ring R over a field F with the same order as G, and returns the normalized unit group of R[G]. It generates a list of normalized units of R[G], constructs a free group F with a generator for each normalized unit, and a list of relations that correspond to the squares of the generators. The unit group of R[G] is then computed as the quotient of F by these relations, and returned as a finitely presented group.

LoadPackage("polycyclic");
F := GF(3);
D_30 := DihedralGroup(30);

FD_30 := GroupRing(F, D_30);

ComputeNormalizedUnitGroup := function(G, R)
local normalized_units, x, el, inv, unit_group, free_group, relations, converted_units;

normalized_units := [];

for x in Elements(G) do
if x <> Identity(G) then
el := One(R) - Embedding(G, R)(x);
inv := One(R) - el;
fi;
od;

normalized_units := Set(normalized_units);
free_group := FreeGroup(Length(normalized_units));

converted_units := List(normalized_units, x -> free_group.1);
relations := List(converted_units, x -> x^2);
unit_group := free_group / relations;

return unit_group;
end;

normalized_unit_group := ComputeNormalizedUnitGroup(D_30, FD_30);
RelatorsOfFpGroup(normalized_unit_group);


gap> LoadPackage("polycyclic");
true
gap> F := GF
(3);
GF(3)
gap> D_30 := DihedralGroup(30);
<pc group of size 30 with 3 generators>
gap>
gap> FD_30 := GroupRing(F, D_30);
<algebra-with-one over GF(3), with 3 generators>
gap>
gap> ComputeNormalizedUnitGroup := function(Gi, R)
>     local normalized_units, x, el, inv, unit_group, free_group, relations, converted_units;
>
>     normalized_units := [];
>
>     for x in Elements(G) do
>         if x <> Identity(G) then
>             el := One(R) - Embedding(G, R)(x);
>             inv := One(R) - el;
>         fi;
>     od;
>
>     normalized_units := Set(normalized_units);
>     free_group := FreeGroup(Length(normalized_units));
>
>     converted_units := List(normalized_units, x -> free_group.1);
>     relations := List(converted_units, x -> x^2);
>     unit_group := free_group / relations;
>
>     return unit_group;
> end;
function( G, R ) ... end
gap>
gap> normalized_unit_group := ComputeNormalizedUnitGroup(D_30, FD_30);
<fp group with 58 generators>
gap> RelatorsOfFpGroup(normalized_unit_group);
[ f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2,
f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2,
f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2, f1^2,
f1^2 ]
gap>

$$$$

• thank you. I will try and tell you . Commented Apr 2, 2023 at 3:00
• What do you mean by "a field F with the same order as G" - that's not the case for $𝐹𝐷_{30}$ above. Commented Apr 3, 2023 at 13:00
• @OlexandrKonovalov i think the same as $G$ means as same as defined above. Commented Apr 4, 2023 at 9:42
• The list of relators does not look right - they are all the same, and why do you require that particular form? Commented Apr 4, 2023 at 23:01