# Finitely generated and FC implies FZ

From Groupprops

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

## 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.