For each quadratic polynomial
f(x) ∈ Z[x], whose ratio of roots is not ±1, a group G(f) of equivalence classes of
certain linear recurrences with companion polynomial f(x) has been constructed by
the author. Its structure was shown to be connected with the structure of the sets of
prime divisors of the linear recurrences. The group G(f) is infinite but its torsion
subgroup is finite and usually, but not always, consists of just two elements; the class
of the Lucas sequence ℐ = [0,1] of f(x) and the class of the recurrence ℰ = [2,P]
associated with f(x). This subgroup is completely determined here for each
polynomial f(x). In 1961 M. Ward raised the question whether (ℐ) and
(ℰ) are the only classes whose sets of prime divisors can be characterized
globally. It is shown in this article that there are groups G(f) with elements of
finite order, other than (ℐ) and (ℰ), whose prime divisors can be similarly
characterized.
