[GAP Forum] About the elements of abstract group SmallGroup(8, 1).

[Max Horn]


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).

Regarding the question below:

> Am 30.04.2022 um 02:20 schrieb Hongyi Zhao <hongyi.zhao at gmail.com>:
> On Fri, Apr 29, 2022 at 10:47 PM Max Horn <horn at mathematik.uni-kl.de> wrote:
>> 
>> Hi,
>> 
>>> Am 28.04.2022 um 13:24 schrieb Hongyi Zhao <hongyi.zhao at 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? 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:

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);
<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