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.k...@iitrpr.ac.in> wrote: > Dear Alexander, Dear forum > > Thank you very much. > > On Thu, Dec 7, 2017 at 7:11 PM, Alexander Konovalov < > alexander.konova...@st-andrews.ac.uk> wrote: > >> >> > On 7 Dec 2017, at 13:19, Surinder Kaur <surinder.k...@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.konova...@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.k...@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* _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
