A group $G$ is called an FC-group if $|x^G|$ is finite for each $x \in G$ (here $x^G$ is the conjugacy class of $x$ in $G$). Equivalently, $|G:C_G(x)|$ is finite for each $x \in G$.
I think it should be completely obvious that a homomorphic image of an FC-group is an FC-group & maybe I'm just tired, but I've been trying to think of a proof of this for a few minutes now and am coming up blank. It's one of these things that is left unproved whenever FC-groups are mentioned, presumably because the proof is straightforward.
Could somebody throw a proof my direction to ease my troubled mind?