FC groups whose periodic parts can be embedded in direct products of finite groups

In this note are considered $FC$ groups whose periodic parts can be embedded in direct products of finite groups. It is shown that if the periodic part of an $FC$ group $G$ can be embedded in the direct product of its finite factor groups with respect to the normal subgroups of $G$ whose intersection is the trivial subgroup, then $G/Z(G)$ is a subgroup of a direct product of finite groups. It is also shown that if the periodic part of an $FC$ group $G$ is a group without a center, then $G$ can be embedded in a direct product of finite groups without centers and a torsion-free Abelian group.