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

[Surinder Kaur]

Dear Alexander, Dear forum

Thank you very much.

On Thu, Dec 7, 2017 at 7:11 PM, Alexander Konovalov <
alexander.konovalov at st-andrews.ac.uk> wrote:

>
> > On 7 Dec 2017, at 13:19, Surinder Kaur <surinder.kaur at iitrpr.ac.in>
> wrote:
> >
> > Dear Forum, Dear Alexander Konovalov,
> >
> > I wanted to calculate the size of the centralizer of an element of V(FG)
> in FG, when F is a finite field with 3 elements and G is a non-abelain
> group of order 3^3. I am unable to do this even with the help of LAGUNA
> package. It is showing that it is "beyond its memory limit."
>
> It's not surprising - you will either run out of memory or run out of time
> if you will try a straightforward approach.
>
> However, you can do efficient calculations of normalisers in the unit group
> given as a pc group:
>
> gap> g:=SmallGroup(3^3,3);;
> gap> f:=GF(3);;
> gap> fg:=GroupRing(f,g);;
> gap> v:=PcNormalizedUnitGroup(fg);
> <pc group of size 2541865828329 with 26 generators>
> gap> s:=Random(v);
> f2^2*f5*f6*f8*f10*f11*f13*f14*f17*f20^2*f24*f25^2
> gap> Centraliser(v,s);
> <pc group of size 4782969 with 14 generators>
>
> and then you only have to deduce how its centraliser in FG looks like.
>
> Hope this helps,
> Alexander
>
>
> > On Mon, Dec 4, 2017 at 3:34 PM, Alexander Konovalov <
> alexander.konovalov at st-andrews.ac.uk> wrote:
> > 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*
> >
> >
> >
> >
> > --
> > Regards
> > Surinder Kaur
> > Research scholar
> > Department of Mathematics
> > IIT Ropar
>
> --
> Dr. Alexander Konovalov, Senior Research Fellow
> Centre for Interdisciplinary Research in Computational Algebra (CIRCA)
> School of Computer Science, University of St Andrews
> Software Sustainability Institute Fellow
> https://alexk.host.cs.st-andrews.ac.uk
> --
> The University of St Andrews is a charity registered in
> Scotland:No.SC013532
>
>
>


-- 
*Regards*
*Surinder Kaur*
*Research scholar  *
*Department of Mathematics *
*IIT Ropar*