Dear Surinder,
No, in this case, LAGUNA can only speed up some calculations
only for those units of this group algebra whose support
consists only of elements whose order is a power of 3.
However, for such small example you can actually do something in GAP:
* construct group algebra
gap> F:=GF(3);
GF(3)
gap> G:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> FG:=GroupRing(F,G);
<algebra-with-one over GF(3), with 2 generators>
* calculate its unit group
gap> U:=Units(FG);
<group with 5 generators>
gap> time;
12484
gap> Size(U);
324
* find normalised units
gap> nu:=Filtered(U,x -> IsOne(Augmentation(x)));;
gap> Length(nu);
162
* construct normalised unit group
gap> V:=Group(nu);
<group with 162 generators>
* construct and explore an isomorphic permutation group
gap> phi:=IsomorphismPermGroup(V);
<action isomorphism>
gap> H:=Image(phi);
<permutation group of size 162 with 162 generators>
gap> IdGroup(H);
[ 162, 41 ]
gap> StructureDescription(H);
"C3 x (((C3 x C3) : C3) : C2)"
* find its minimal generating set and map it back to the group algebra
gap> mgs:=MinimalGeneratingSet(H);;
gap> List(mgs,u->PreImagesRepresentative(phi,u));
[ (Z(3))*()+(Z(3)^0)*(2,3)+(Z(3))*(1,2)+(Z(3)^0)*(1,2,3)+(Z(3)^0)*(1,3,2),
(Z(3))*()+(Z(3))*(2,3)+(Z(3))*(1,2)+(Z(3))*(1,2,3)+(Z(3))*(1,3,2),
(Z(3)^0)*()+(Z(3)^0)*(1,2)+(Z(3))*
(1,2,3)+(Z(3)^0)*(1,3,2)+(Z(3))*(1,3) ]
Hope this helps
Alexander
> On 4 Mar 2017, at 06:38, Surinder Kaur <[email protected]> wrote:
>
> Using LAGUNA package we can calculate the Normalized unit group of a
> p-modular group algebra but can we find it if group algebra is not
> p-modular like of (GF(3)S3) where S3 is symmetric group of order 6.
>
>
> --
> *Regards*
> *Surinder Kaur*
> *Research scholar *
> *Department of Mathematics *
> *IIT Ropar*
_______________________________________________
Forum mailing list
[email protected]
http://mail.gap-system.org/mailman/listinfo/forum
