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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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.