## Transitivity-related property of finite permutation groups

Let $\cal F$ denote the group of all finitely-supported permutations of $\mathbb N$. Say that a finite subgroup $G$ of $\cal F$ is singular if $G$ acts transitively on $\lbrace 1,2,3 \rbrace$ but no cyclic subgroup of $G$ acts transitively on $\lbrace 1,2,3 \rbrace$ (this is equivalent to saying that some element in $G$ sends $1$ to $2$, another sends $1$ to $3$ but no element of $G$ has all of $1,2$ and $3$ in a single orbit).

The Klein group (products of disjoint transpositions on $\lbrace 1,2,3,4 \rbrace$) is in example of such a subgroup.

Question 1 : are there other simple examples of minimal singular subgroups ? Is there a parametric description of all of them up to isomorphism ?

Question 2 : Denote by ${\cal F}(i \to j)$ the set of all permutations in $\cal F$ sending $i$ to $j$. Say that a permutation $s\in {\cal F}(1 \to 2)$ and a permutation $t\in {\cal F}(1 \to 3)$ are related iff the subgroup generated by $s$ and $t$ is a minimal singular subgroup of $\cal F$. Given $s$, let $R(s)$ denoted the set of all $t$'s such that $s$ and $t$ are related. Does $R(s)$ admit a simple description ?

Of course, any answer to question 2 automatically provides an answer to question 1.

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A simple example for Q1 generalizing your example for the Klein group:

Let $G$ be any non-cyclic finite group generated by two elements $g$ and $h$. Take the regular action of $G$ on itself, i.e., let $G$ act on itself by right rsp. left multiplication. Identify $G$ (as set acted upon) with a subset of $\mathbb N$ where the $1$ of the group is identified with $1 \in \mathbb N$, $g$ is identified with $2$ and $h$ with $3$.

As the orbits of the cyclic subgroups of $G$ containing $1$ are just the cyclic subgroups, none of them containing both $g$ and $h$, $G$ considered as subgroup (the regular action is faithful) of $\cal F$ is minimal singular.

Probably the condition minimal singular is too weak to be helpful. By the way, "finitely-supported" permutations are often called finitary.

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