[GAP Forum] All factorizations of a permutation

[Max Horn]

Dear Ahmet,

> On 23. Apr 2021, at 12:32, Ahmet Arıkan <arikan at gazi.edu.tr> wrote:
> 
> Thank you Max,
> 
> I think I can clarify more as follows:
> 
> My initial question

I didn't see this initial question, was that in another email or just not (yet) on the mailing list?

> “Is there a perfect p-group G contained FSym(IN*) which satisfies my condition (G has a generating subset X such that every infinite subset of X also generates G—-I call such groups “infinitely supported”). Here IN* is the set of non-zero natural numbers and FSym(IN*) is the group of all finitary permutations of IN* as usual. It can be found some examples for non-perfect or simple cases, but such cases are not under my consideration for now.

Then I guess I am still missing something and/or was mistaken in my previous email -- because I would have thought the argument from my previous email shows that no subgroups of FSym(|N*) with a generating set X as demanded can exist (perfect or not, doesn't matter). Yet you say there are known examples. Where is the mistake in my argument? Perhaps you can tell us one of these examples?


Best wishes
Max



> 
> To know the existence of such perfect groups shall provide a very important point of view for some other crucial problems. In my opinion, we need to start with some (useful?) elements to construct such an X using GAP. So I need the distinct factorizations of elements to construct X (I guess!).
> 
> Best wishes,
> Ahmet Arikan
> 
> 
> 
> ——————————-
> Gazi University, 
> Gazi Education Faculty, 
> Department of Mathematics Education,
> Ankara, 
> Turkey
> 
> Max Horn <horn at mathematik.uni-kl.de> şunları yazdı (23 Nis 2021 12:35):
> 
>> 
>> 
>>> On 23. Apr 2021, at 10:25, Ahmet Arıkan <arikan at gazi.edu.tr> wrote:
>>> 
>>> Hi Chris, thank you for the reply. 
>>> 
>>> My main problem is to construct a group G in Sym(\Omega) satisfying the following property: 
>> 
>> What is \Omega? An arbitrary infinite set?
>> 
>>> 
>>> G has a generating subset X such that every infinite subset of X also generates G.
>>> 
>>> So to construct such a group, we may start with an element x (say x=(1,3,4) ) to contruct X. Then we need to find suitable factorizations like (1,3,4)=(1,2,3,4)*(2,3) ( or multiple factorizations)  and  continue to construct X={(1,3,4), (1,2,3,4),(2,3),...}. This is just an explanation of why I want to find suitable factorizations of permutations.
>> 
>> I don't see at all why you "may start" wich such elements. To the contrary, I think looking at such elements won't help at all.
>> 
>> For suppose G and X are as desired. Then G must be non-trivial and hence there is a point a \in \Omega which G moves. Let 
>> 
>>  X' := { \pi \in X |  a^\pi \neq a }
>> 
>> Then this set still must be infinite, for if it was finite, then X'':=X\setminus X' would be an infinite subset of X but the group it generates fixes a and hence cannot be G.
>> 
>> So we may replace X by X'. Now we can repeat this process for any point moved by G (of which there must be infinitely many).
>> 
>> In the end, the set X only contains permutations with infinite support. Moreover, we can of course restrict it to be countably infinite.
>> 
>> If I am not mistaken, here is an example for a group and set as described: G=\Sym(\ZZ) together with 
>> 
>> X := \{ \pi_p | p is a prime \}
>> 
>> and
>> 
>> \pi_p: ZZ\to\ZZ, x \mapsto x + p
>> 
>> It satisfies the even strong property that any subset of X of size at least 2 still generates G.
>> 
>>> We do not know yet if such a perfect locally finite (p-) group G exists.
>> 
>> Why do you think such a group must be perfect? Or are you asking whether such a group *can* be perfect? The example I gave of course is abelian and not locally finite. So perhaps there are more requirements in your question that are missing?
>> 
>> 
>> 
>> 
>> Cheers
>> Max
>>