[GAP Forum] Information about p-modular group algebra FG in GAP

[Alexander Konovalov]

Dear Surinder,

You have 3^27 elements in fg, and 3^26 of them of augmentation one, so the calculation
which you're trying to perform is not feasible. You need to use the LAGUNA package
to be able work with normalised unit group of fg in a very efficient pc presentation 
and then interpret the result in terms of fg. See, for example, a sample calculation
at https://gap-packages.github.io/laguna/doc/chap2.html

For example, in your setup, you can find the minimal generating set of the 
normalised unit group as follows:

gap> g:=SmallGroup(3^3,3);;
gap> f:=GF(3);;
gap> fg:=GroupRing(f,g);;
gap> u:=NormalizedUnitGroup(fg);
<group of size 2541865828329 with 26 generators>
gap> v:=PcNormalizedUnitGroup(fg);
<pc group of size 2541865828329 with 26 generators>
gap> MinimalGeneratingSet(v);
[ f1, f2, f3, f4, f6, f7, f9, f12, f21, f26 ]
gap> gens:=MinimalGeneratingSet(v);
[ f1, f2, f3, f4, f6, f7, f9, f12, f21, f26 ]
gap> phi:=NaturalBijectionToNormalizedUnitGroup(fg);;
gap> List(gens,x -> x^phi);
[ (Z(3)^0)*f1, (Z(3)^0)*f2, (Z(3))*<identity> of ...+(Z(3)^0)*f2+(Z(3)^0)*f2^
    2, (Z(3))*<identity> of ...+(Z(3))*f1+(Z(3))*f2+(Z(3)^0)*f1*f2, 
  (Z(3))*<identity> of ...+(Z(3)^0)*f1+(Z(3)^0)*f1^2, 
  (Z(3)^0)*f1+(Z(3))*f2+(Z(3)^0)*f1*f2+(Z(3))*f2^2+(Z(3)^0)*f1*f2^2, 
  (Z(3))*f1+(Z(3)^0)*f2+(Z(3))*f1^2+(Z(3)^0)*f1*f2+(Z(3)^0)*f1^2*f2, 
  (Z(3))*<identity> of ...+(Z(3)^0)*f1+(Z(3)^0)*f2+(Z(3)^0)*f1^2+(Z(3)^
    0)*f1*f2+(Z(3)^0)*f2^2+(Z(3)^0)*f1^2*f2+(Z(3)^0)*f1*f2^2+(Z(3)^0)*f1^
    2*f2^2, (Z(3))*f1+(Z(3))*f2+(Z(3)^0)*f3+(Z(3))*f1^2+(Z(3))*f1*f2+(Z(3)^
    0)*f1*f3+(Z(3))*f2^2+(Z(3)^0)*f2*f3+(Z(3))*f1^2*f2+(Z(3)^0)*f1^2*f3+(
    Z(3))*f1*f2^2+(Z(3)^0)*f1*f2*f3+(Z(3)^0)*f2^2*f3+(Z(3))*f1^2*f2^2+(Z(3)^
    0)*f1^2*f2*f3+(Z(3)^0)*f1*f2^2*f3+(Z(3)^0)*f1^2*f2^2*f3, 
  (Z(3))*<identity> of ...+(Z(3)^0)*f1+(Z(3)^0)*f2+(Z(3)^0)*f3+(Z(3)^0)*f1^
    2+(Z(3)^0)*f1*f2+(Z(3)^0)*f1*f3+(Z(3)^0)*f2^2+(Z(3)^0)*f2*f3+(Z(3)^0)*f3^
    2+(Z(3)^0)*f1^2*f2+(Z(3)^0)*f1^2*f3+(Z(3)^0)*f1*f2^2+(Z(3)^0)*f1*f2*f3+(
    Z(3)^0)*f1*f3^2+(Z(3)^0)*f2^2*f3+(Z(3)^0)*f2*f3^2+(Z(3)^0)*f1^2*f2^2+(
    Z(3)^0)*f1^2*f2*f3+(Z(3)^0)*f1^2*f3^2+(Z(3)^0)*f1*f2^2*f3+(Z(3)^
    0)*f1*f2*f3^2+(Z(3)^0)*f2^2*f3^2+(Z(3)^0)*f1^2*f2^2*f3+(Z(3)^0)*f1^
    2*f2*f3^2+(Z(3)^0)*f1*f2^2*f3^2+(Z(3)^0)*f1^2*f2^2*f3^2 ]

Please do not hesitate ask me if you will have further questions.

Best regards,
Alexander


> On 4 Dec 2017, at 05:47, Surinder Kaur <surinder.kaur at iitrpr.ac.in> wrote:
> 
> Dear Forum
> 
> I wanted to get some information in GAP about the elements of augmentation
> 1 in the group algebra FG, where F is a Galois field with 3 elements and G
> is non-abelian of order 3^3.
> 
> I am trying this way:
> 
> g:=SmallGroup(3^3,3);;
> f:=GF(3);;
> fg:=GroupRing(f,g);;
> e:=Identity(fg);;
> m:=MinimalGeneratingSet(g);;
> v:=Filtered(fg,x->Augmentation(x) = Z(3)^0);;
> Print (v[1], "\n");
> 
> 
> But I am getting that "it has reached pre-set memory limit".
> 
> How can I get the elements of v. Any suggestion will be highly appreciated.
> 
> -- 
> 
> *Regards**Surinder Kaur*
> *Research scholar  *
> *Department of Mathematics *
> *IIT Ropar*