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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3All 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 05 '12 at 17:58
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For a proof that subgroups of finite index in a finitely generated group are finitely generated, see here. – Arturo Magidin Jul 05 '12 at 18:16
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Thank you both very much, i got it now! – Boris Jul 05 '12 at 22:05
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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")
Arturo Magidin
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