  A nontrivial element in a group is a generalized torsion element if some
nonempty finite product of its conjugates is the identity. We prove that any
generalized torsion element in a free product of torsion-free groups is
conjugate to a generalized torsion element in some factor group. This implies
that the fundamental group of a compact orientable $3$-manifold $M$ has a
generalized torsion element if and only if the fundamental group of some prime
factor of $M$ has a generalized torsion element. On the other hand, we
demonstrate that there are infinitely many toroidal $3$-manifolds whose
fundamental group has a generalized torsion element, while the fundamental
group of each decomposing piece has no such elements.
