> On 23. Apr 2021, at 10:25, Ahmet Arıkan <ari...@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
_______________________________________________
Forum mailing list
Forum@gap-system.org
https://mail.gap-system.org/mailman/listinfo/forum
