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

[Alexander Konovalov]

> 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
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