# Center of a finitely generated fc group

is the center of a finitely generated fc group (a group in which every conjugacy class is finite) also finitely generated? And if yes, how can I prove it?

• All centralizers have finite index, so the intersection of the centralizers of a finite generating set, which is equal to the center, also has finite index and hence is finitely generated. – Derek Holt Jul 5 '12 at 17:58
• For a proof that subgroups of finite index in a finitely generated group are finitely generated, see here. – Arturo Magidin Jul 5 '12 at 18:16
• Thank you both very much, i got it now! – Boris Jul 5 '12 at 22:05

As Derek Holt notes in the comment, the center of a finitely generated FC-group is always of finite index in the group: the centralizer of any element is of finite index (the index equals the cardinality of the conjugacy class), and the center is the intersection of the centralizers of a generating set. The intersection of finitely many subgroups of finite index is itself of finite index, thus showing that $[G:Z(G)]\lt\infty$.
If $G$ is finitely generated and $H$ is a subgroup of finite index, then $H$ is finitely generated.