Finitely generated and FC implies FZ

This article gives a proof/explanation of the equivalence of multiple definitions for the term finitely generated FZ-group
View a complete list of pages giving proofs of equivalence of definitions

Statement

Suppose $G$ is a finitely generated group that is a FC-group: every conjugacy class in $G$ is finite. Then, $G$ is a FZ-group: the center of $G$ has finite index in $G$.

Proof

The proof idea is that the center is the intersection of centralizers of elements in a generating set. This is an intersection of finitely many subgroups of finite index, hence is finite. Details to be filled in.