Finitely generated and FC implies FZ

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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.