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Have you any nice example of central divisible subgroup of a finitely presented group ? (of course the subgroup has not to be trivial)

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This depends on what you understand by a "nice example". Have you read the discussion on MathOverflow: ? – Derek Holt Jul 1 '12 at 16:58
I read this discussion and I hoped it was simpler for another divisible group than Q. A nice example would be a well-known group or a presentation with few generators and relators. – Seirios Jul 1 '12 at 21:32

I found a weak solution of my problem: Let $A_n$ denote the Abel group. For $n \geq 4$, $A_n$ is finitely presented and $Z(A_n) \simeq Z[1/p]$ is $p$-divisible.

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