Is the center of an infinite solvable group infinite?

Question

Is the center of every infinite solvable group infinite? I can see why this must be true for solvable groups constructed by taking semidirect products, but I don't think all infinite solvable groups come from iterated semidirect products of abelian groups with other groups. Neither can I find any counterexamples of the claim. I'd be much obliged if someone could hint at why this claim must be true, or provide a counterexample. Thanks.

Answer 1

Since nilpotent groups are solvable, we can also look for infinite nilpotent groups with finite center. These groups cannot be finitely generated as all finitely generated nilpotent groups with finite center are finite. One example should be given by taking the infinite central product of the quaternion group $Q_8$. Then the center is cyclic of order $n = 2$.