Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the *class
transposition* $\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$
which interchanges $r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$
and which fixes everything else.

> <b>Question:</b> Let $G < {\rm Sym}(\mathbb{Z})$ be a group generated by
> $3$ class transpositions, and assume that the integers $0, \dots, 42$
> all lie in the same orbit under the action of $G$ on $\mathbb{Z}$.
> Is the action of $G$ on $\mathbb{N}_0$ necessarily transitive?

*Remarks:*

  - When replacing $42$ by $41$, the answer obviously gets negative since
    the finite group
    $$
      G \ := \ \langle \tau_{0(2),1(2)}, \tau_{0(3),2(3)}, \tau_{0(7),6(7)} \rangle
    $$
    acts transitively on the set $\{0, \dots, 41\}$.
    Therefore if true, the assertion is sharp.

  - There is computational evidence suggesting that there is, say,
    "a reasonable chance" that the answer is positive.

  - A positive answer would mean that groups generated by $3$ class
    transpositions are "well-behaved" in the sense that for deciding
    transitivity, looking at very small numbers is sufficient, and that
    for larger numbers "nothing can happen any more".

    *Added on Jun 20, 2015:* A positive answer would however *not* imply
    that all questions on groups generated by $3$ class transpositions
    are algorithmically decidable. 

  - A positive answer would imply the [Collatz conjecture][2].
    On the other hand, if the Collatz conjecture holds, this would (by far!)
    not imply a positive answer to the question.

    *Added on Jun 20, 2015:* The reason why a positive answer would imply the
    Collatz conjecture is that the group
    $$
      C := \langle \tau_{0(2),1(2)}, \tau_{1(2),2(4)}, \tau_{1(4),2(6)} \rangle
    $$
    acts transitively on $\mathbb{N}_0$ if and only if the Collatz conjecture holds.

  - There is a related question [here][1].

**Added on Jun 20, 2015:**

  - Example of a group which does act transitively: the group
    $$
      G := \langle \tau_{0(2),1(2)}, \tau_{0(3),2(3)}, \tau_{1(2),2(4)} \rangle
    $$
    acts at least $5$-transitively on $\mathbb{N}_0$.

  - Example of an infinite group $G$ such that the numbers $0, \dots, 25$ all lie in
    the same orbit under the action of $G$ on $\mathbb{Z}$, but which *likely*
    does not act transitively on $\mathbb{N}_0$:
    $$
      G := \langle \tau_{0(2),1(2)}, \tau_{0(2),1(4)}, \tau_{0(6),5(9)} \rangle
    $$

  - *(Easy case.)* The answer is positive for groups generated by $3$
    class transpositions which interchange residue classes *with the same moduli*
    (this is the case where no multiplications and no divisions occur).
    Transitivity on $\mathbb{N}_0$ obviously cannot occur in this case.
    More precisely, if we have a group generated by $k$ such class transpositions,
    the length of an orbit is bounded above by $a_k$, where $a_0 = 1$ and
    $a_{k+1} = a_k \cdot (a_k + 1)$.

  - Since HJRW suggested to look for "undecidability phenomena":
    so far I don't know any for groups generated by $3$ class transpositions, but there
    are groups generated by $4$ class transpositions which have finitely generated
    subgroups with unsolvable membership problem.

    For example, putting $\kappa := \tau_{0(2),1(2)}$, $\lambda := \tau_{1(2),2(4)}$,
    $\mu := \tau_{0(2),1(4)}$ and $\nu := \tau_{1(4),2(4)}$, the group
    $V := \langle \kappa, \lambda, \mu, \nu \rangle$
    is isomorphic to [Thompson's group V][3]. Since the free group of rank $2$ and
    $V \times V$ both embed into $V$, it follows from a result of Mihailova that $V$ has
    subgroups with unsolvable membership problem.

    Side remark: $V$ is actually also the group generated by all class transpositions
    interchanging residue classes modulo powers of $2$; in general, groups may get *a lot*
    more complicated once the moduli of the residue classes interchanged by the
    generators are not all powers of the same prime.

**Update of Nov 10, 2016:**

Unfortunately the answer to the question as it stands turned out
to be negative. -- These days I found a counterexample: put
$$
  G := \langle \tau_{0(2),1(2)}, \tau_{0(2),3(4)}, \tau_{4(9),2(15)} \rangle.
$$
Then all integers $0, 1, \dots, 87$ lie in one orbit under the action
of $G$ on $\mathbb{Z}$, but $G$ is not transitive on $\mathbb{N}_0$
since $88$ lies in another orbit.

The crucial feature of this example appears to be that intransitivity
is forced by the existence of a nontrivial partition of $\mathbb{Z}$
into unions of residue classes modulo $180$ which $G$ stabilizes setwisely.
The modulus $180$ happens to be the least common multiple of the
moduli of the residue classes interchanged by the generators of $G$.

This suggests to reformulate the question as follows:

> **Question (new version):** Let $G < {\rm Sym}(\mathbb{Z})$ be
> a group generated by $3$ class transpositions, and let $m$ be
> the least common multiple of the moduli of the residue classes
> interchanged by the generators of $G$. Assume that $G$ does not
> setwisely stabilize any union of residue classes modulo $m$
> except for $\emptyset$ and $\mathbb{Z}$, and assume that the
> integers $0, \dots, 42$ all lie in the same orbit under the action
> of $G$ on $\mathbb{Z}$. Is the action of $G$ on $\mathbb{N}_0$
> necessarily transitive?

*Remarks:*

  - If true, the assertion is still sharp in the sense that the
    bound $42$ cannot be replaced by $41$ (cf. the first remark
    on the original question).

  - It is conceivable that the assertion needs to be further weakened
    a little by assuming that $G$ does not setwisely stabilize any
    union of residue classes except for $\emptyset$ and $\mathbb{Z}$.
    (Also in this case a positive answer to the question would
    still imply the Collatz conjecture.)

  [1]: https://mathoverflow.net/questions/112527/groups-generated-by-3-involutions
  [2]: https://en.wikipedia.org/wiki/Collatz_conjecture
  [3]: https://en.wikipedia.org/wiki/Thompson_groups