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Does there exist an infinite nilpotent group with finite center?

I failed to prove that it does not, but any examples of such groups do not come to my mind either.

Any help will be appreciated.

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Yes, but such examples cannot be finitely generated.

An infinite extraspecial group is an example. Let $p$ by a prime, and let $G$ be the group defined by the presentation $$\langle x_i\ (i \in {\mathbb Z} \setminus \{0\}),z \mid x_i^p=z^p = [x_i,z] = [x_i,x_j]=1\ (|i| \ne |j|),\,[x_i,x_{-i}]=z \rangle.$$ Then $Z(G) = \langle z \rangle$ has order $p$.

This group could also be described as the central product of infinitely many copies of a nonabelian group of order $p^3$.

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