  Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty
finite product of conjugates of $g$ equals to the identity, then $g$ is called
a generalized torsion element. The minimum number of conjugates in such a
product is called the order of $g$. We will classify $3$-manifolds $M$, each of
whose fundamental group has a generalized torsion element of order two.
Furthermore, we will classify such elements in $\pi_1(M)$. We also prove that
$R$-group and $\overline{R}$-group coincide for $3$-manifold groups, and
classify $3$-manifold groups which are $R$-groups (and hence
$\overline{R}$-groups).
