Let $G$ be a group such that its center $Z(G)$ has finite index in $G$. Show that every conjugation class has finite elements.
I don't know how to attack the problem. I thought the following: if $C$ is a conjugacy class, then $C$ can be thought as an orbit under the action of $G$ on itself by conjugation. Could it be that the orbit divides $[G:Z(G)]$?, I am not so sure if this is true, if this was the case then from here it follows the statement of the problem. I would appreciate hints or ideas to solve the problem.