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

Fri Dec 29 20:37:40 GMT 2017

Dear Forum,

If FG is the group algebra of the Dihedral group G of order 6 over the
finite field with 9(=3^2) elements, then how can we can the normalized unit
group of FG be obtained. When I take the field with 3 elements, then I am
able to get the elements, but if I take the field with 9 elements then what
should be the approach. Any suggestions will be highly appreciated.

g:=DihedralGroup(6);;
f:=GF(3);;
fg:=GroupRing(f,g);;
e:=Identity(fg);;
m:=MinimalGeneratingSet(g);;
l:=List(m,x-> x^Embedding(g,fg));;
u:=Units(fg);;
s:=Filtered(u, x-> Augmentation(x) = (Z(3)^(0)) );;
v:=AsGroup(s);;
Print(v);

This was the approach I used for group algebra of smaller order, but it
didn't work anymore when I increased the size of field.

On Sat, Dec 9, 2017 at 4:50 PM, Surinder Kaur <surinder.kaur at iitrpr.ac.in>
wrote:

> 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)*f
>> 3+(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*f
>> 3+(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*
>

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