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.


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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