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?
Thanks in advance
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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 $[G:Z(G)]\lt\infty$.
Now the result comes down to the following:
If $G$ is finitely generated and $H$ is a subgroup of finite index, then $H$ is finitely generated.
There are three proofs of this result in this previous question.
(CW, since Prof Holt got there first, but hoping this will prevent the question from being "unanswered")