18.47 (2014)

Is it algorithmically decidable whether a group generated by three given class transpositions (for the definition, see 17.57)
$\qquad$ a) has only finite orbits on $\mathbb{Z}$?
$\qquad$ b) acts transitively on the set of nonnegative integers in its support?
A difficult case is the group $\langle \tau_{1(2), 4(6)}, \tau_{1(3), 2(6)}, \tau_{2(3), 4(6)} \rangle$, which acts transitively on $\mathbb{N} \setminus 0(6)$ if and only if Collatz’ $3n + 1$ conjecture is true.