transitive subgroup of an action

Question

Suppose that we have a group G acting transitively on the set X, is there a smallest subgroup H of G, such that H acts transitively on X?

I was thinking if it's possible to construct H, such that for any x and y in X, there is a unique h in H that maps from x to y.

Answer 1

There is not necessarily a unique smallest transitive subgroup. A simple example of that is$H$ dihedral of order $8$ acting on $4$ points, for which there are two (necessarily minimal) transitive subgroups of order $4$, and they are both regular, so they both have the uniqueness property you mentioned.

As for whether there exists some minimal transitive subgroup, clearly yes for finite $X$, but probably not in general.

There are finite examples with no regular subgroups. One such example is $A_4$ acting on $6$ points. It is well-known that $A_4$ has no subgroup of order $6$.

Answer 2

Take the additive $\mathbb{Z}$, acting naturally on $\mathbb{Z}_2 = \{0,1\}$.

Take any odd number $n$.

Then, $n\mathbb{Z}$ acts transitively over $\mathbb{Z}_2$. This is a descending chain of subgroups of $\mathbb{Z}$ acting transitively on $\mathbb{Z}_2$. Their intersection is $\{0\}$, which is not transitive.

If $n$ is even, then $n\mathbb{Z}$ does not act transitively.

So, there is no minimal subgroup for this action.