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

  • 3
    $\begingroup$ 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. $\endgroup$ – Derek Holt Jul 5 '12 at 17:58
  • $\begingroup$ For a proof that subgroups of finite index in a finitely generated group are finitely generated, see here. $\endgroup$ – Arturo Magidin Jul 5 '12 at 18:16
  • $\begingroup$ Thank you both very much, i got it now! $\endgroup$ – 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 $[G:Z(G)]\lt\infty$.

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")


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