[GAP Forum] All factorizations of a permutation

[Max Horn]

> 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