On Sat, Apr 30, 2022 at 7:39 PM Max Horn <h...@mathematik.uni-kl.de> wrote: > > > Hi Hongyi, > > to you and all others involved in these discussions: Apologies for being a > bit blunt, but: While it's great that you have lots of questions and helpful > answers, I'd appreciate if y'all either kept the discussion to the list (my > personal preference: this way we avoid having multiple people spending time > answering the same question again and again; and others can benefit from the > answers; and finally, it increases the chance of getting an answer, and it > also allows others to comment and improve on answers), or else leave it > completely off the list (as seeing half the emails from a conversation is > rather confusing).
Do you mean that I shouldn't have deleted any content from the original message which I reply on? My habit is only based on the habits that I see on other technical lists. For example: For a reply, if a lot of content has nothing to do with what I reply, I may delete those. I don't know what is the usual practice used by the pure and applied mathematics lists like GAP. > Regarding the question below: > > > Am 30.04.2022 um 02:20 schrieb Hongyi Zhao <hongyi.z...@gmail.com>: > > On Fri, Apr 29, 2022 at 10:47 PM Max Horn <h...@mathematik.uni-kl.de> > > wrote: > >> > >> Hi, > >> > >>> Am 28.04.2022 um 13:24 schrieb Hongyi Zhao <hongyi.z...@gmail.com>: > >>> Hi GAP team, > >>> For the abstract group SmallGroup(8,1), I noticed the following > >>> related information: > >>> gap> g:=SmallGroup(8,1); > >>> <pc group of size 8 with 3 generators> > >>> gap> Elements(g); > >>> [ <identity> of ..., f1, f2, f3, f1*f2, f1*f3, f2*f3, f1*f2*f3 ] > >>> gap> SmallGeneratingSet(g); > >>> [ f1 ] > >>> gap> StructureDescription(g); > >>> "C8" > >>> As you can see, it's a cyclic group of order 8. So, I wonder why the > >>> elements don't look like this: > >>> [ <identity> of ..., f1, f1^2, f1^3, f1^4, f1^5, f1^6, f1^7 ] > >> > >> There are infinitely many ways to describe a group of order 8. In this > >> case, GAP chooses to represent it as so-called pc-group with a "refined > >> pc-series", which happens to have three generators. However, the first one > >> actually generates the group. That is: > >> > >> gap> Order(g.1) > >> 8 > >> > >> If you for some reason want a group where the names of the generators are > >> like what you wrote, you could use an "fp group" (finitely presented group) > >> > >> gap> g:=CyclicGroup(IsFpGroup,8); > >> <fp group of size 8 on the generators [ a ]> > >> gap> Elements(g); > >> [ <identity ...>, a, a^2, a^3, a^4, a^5, a^6, a^7 ] > > > > There are five groups of order 8 in total. How can I use a method > > similar to the above to represent the other four? > > I am not sure what the question is. Are you asking for a finite presentation > of the groups? Exactly. > For any (finite) group G, you can get an isomorphic "fp group" (i.e. a group > given by a finite presentation) via the command IsomorphismFpGroup. Here's an > extended example: Thank you for your excellent explanation and example. > gap> G:=SmallGroup(8,3); > <pc group of size 8 with 3 generators> > gap> iso:=IsomorphismFpGroup(G); > [ f1, f2, f3 ] -> [ F1, F2, F3 ] > gap> H:=Image(iso); Why must I first use the Image command to create another group H, and cannot use iso directly for the subsequent testing? > <fp group of size 8 on the generators [ F1, F2, F3 ]> > gap> RelatorsOfFpGroup(H); > [ F1^2, F2^-1*F1^-1*F2*F1*F3^-1, F3^-1*F1^-1*F3*F1, F2^2, F3^-1*F2^-1*F3*F2, > F3^2 ] > gap> H2:=SimplifiedFpGroup(H); > <fp group of size 8 on the generators [ F1, F2 ]> > gap> RelatorsOfFpGroup(H2); > [ F1^2, F2^2, (F2*F1)^4 ] > > > Cheers > Max Regards, Hongyi _______________________________________________ Forum mailing list Forum@gap-system.org https://mail.gap-system.org/mailman/listinfo/forum
