[GAP Forum] LAGUNA package

[Alexander Konovalov]

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 <surinder.kaur at iitrpr.ac.in> 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*